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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I ^1 I HYDRODYNAMICS CAMBRIDGE UNIVERSITY PRESS C F. CLAY, Manager ftonlron: FETTER LANE, E.C. eHinllttrglft : xoo PRINCES STREET lUiD «orli: G. P. PUTNAM'S SONS ISomftat, Calcutta anti iKaDTas: MACMILLAN AND CO. ,1 Ltd. Corotttu: J. M. DENT AND SONS, Ltd. ffoitso: THE MARUZENKABUSHIKIKAISHA Ail rights teservifd >f*^ ^ HYDRODYNAMICS BY HORACE LAMB, M.A., LL.D., Sc.D., F.R.S. PROFESSOR OF MATHEMATICS IN THE VICTORIA UNIVERSITY OF MANCHESTER; FORMERLY FELLOW OF TRINITY COLLEGE, CAMBRIDGE FOURTH EDITION « > • • • "4 • • • • • . Cambridge : at the University Press 1916 First EdiUan 1879 Second Edition 1896 Third EdiUon 1906 Fourth Edition 1916 * \ I TO HENRY MARTYN TAYLOR THIS BOOK IS INSCRIBED BY HIS FRIEND THE AUTHOR PREFACE rilHIS book may be regarded as a fourth edition of a Treatise on the "*- Mathematical Theory of the Motion of Fluids^ which was published in 1879. The second edition, largely remodelled and extended, appeared under the present title in 1895, and was followed by a third in 1906. In this issue, as in the preceding one, no change has been made in the general plan and arrangement, but the work has been carefully revised, occasional passages have been rewritten, and various interpolations and additions have been made, more especially in the latter part of the book, which deals mainly with physical applications. A few investigations of secondary interest have been condensed or omitted. A word or two may be said with regard to certain departures from general usage which are to be found in the book. The use of the reversed sign for the velocity-potential {<f>), which was adopted in the 1895 edition and is here continued, was not altogether an innovation, and has strong arguments of a physical kind to recommend it. It appears so much more natural to regard the state of motion of a dynamical system, in any given configuration, as specified by the impulses which would start it, rather than by those which would stop it, that the altered definition of the function would seem to require no further justification. It has also the advantage that the analogies with other branches of Mathematical Physics are rendered more complete. In the present edition I have been led to make a further slight change from prevalent usage, by choosing for special designation the vector (|, rq, t) whose components in terms of the velocity (w, v, w) are dw dv du dw dv du dy dz' dz dx* dx dy ' rather than that whose components have half these values. This procedure avoids the insertion of an unnecessary factor 2 or J in a number of formulae, in particular in the theorem of Stokes (Art. 32) which is the fundamental BEQUEST OF ALEXANDER ZIWIT« Preface vii relation in the present connection. The vector (^, r), ^) as now defined may conveniently be called the *vorticity,' the term * rotation' being used, if required, in its established sense. It may be added that the altered notation is in conformity with the physical analogies already referred to. It is more- over already current in some writings on the present subject. Pains have been taken to make due acknowledgment of authorities in the footnotes ; but it will be understood that the original methods have not always been followed in the text. I would add that the work has less pretensions than ever to be regarded as a complete account of the science with which it deals. The subject has of late attracted increased attention in various countries, and it has become correspondingly difficult to do justice to the growing literature. Some memoirs deal chiefly with questions of mathematical method and so fall outside the scope of this book; others though physically important hardly admit of a condensed analysis; others, again, owing to the multiplicity of publications, may imfortunately have been overlooked. And there is, I am afraid, the inevitable personal equation of the author, which leads him to take a greater interest in some branches of the subject than in others. I am much indebted to the staff of the University Press for their careful supervision of the printing, and for kindly calling attention to various over- sights. It is again a satisfaction to me to inscribe on the fly-leaf the name of Mr H. M. Taylor, whose kindly encouragement first led me to write on the subject, and whose help in revision I had gratefully to acknowledge on former occasions. HORACE LAMB. January, 1010. CONTENTS ABT. 1. 2. 3>9. 10. 11. 12. 13, 14. 16. 16. 17. 18, 19. 20. 21-23. 24. 26. 26-29. 30. 31, 32. 33. CHAPTER I THE EQUATIONS OF MOTION PAGE Fundamental property of a fluid 1 The two plans of investigation 2 *Eulerian' form of the equations of motion. Dynamical equations. Equation of continuity. Physical equations. Surface conditions 2 Equation of energy 8 Impulsive generation of motion 11 Equations referred to moving axes 12 'Lagrangian' form of the dynamical equations and of the equation of continuity 13 Weber's transformation 14 Extension of the Lagrangian notation 16 CHAPTER II INTEGRATION OP THE EQUATIONS IN SPECIAL CASES Velocity-potential. Lagrange's theorem 16 Physical and kinematical relations oi<f> 17 Integration of the equations when a velocity-potential exists; pressure- equation 18 Steady motion. Deduction of the pressure-equation £rom the principle of energy. Limit to the velocity 19 Efflux of liquids; vena contracta 22 Efflux of gases 24 Examples of rotating fluid. Uniform rotation. Bankine's * combined vortex.' Electromagnetic rotation 26 CHAPTER III IRROTATIONAL MOTION Analysis of the differential motion of a fluid element into strain and rotation 29 *Flow' and * circulation.' Stokes' theorem 31 Constancy of circulation in a moving circuit 34 Contents IX ABT. PAGE 34, 35. Irrotational motion in simply-connected spaces; single- valued velocity- potential 35 36-30. Incompressible fluids; tubes of flow. <f> cannot be a maximum or minimum. The velocity cannot be a maximum. Mean value of <f> over a spherical surface 36 40, 41. Conditions of determinateness of ^ 39 42-46. Green's theorem ; dynamical interpretation; formula for kinetic energy. Kelvin's theorem of minimum energy 42 47, 48. Multiply-connected regions; 'circuits' and 'barriers' .... 47 49-51. Irrotational motion in multiply-connected spaces ; many-valued velocity potential; cyclic constants 48 52. Case of incompressible fluids. Conditions of determinateness of ^ . . 51 53-55. Kelvin's extension of Green's theorem; dynamical interpretation; energy of an irrotationally moving liquid in a cyclic space . 52 56-58. 'Sources' and 'sinks.' Double sources. Irrotational motion of a liquid in terms of surface-distributions of sources 55 CHAPTEE IV MOTION OP A LIQUID IN TWO DIMENSIONS 59. Lagrange's stream-function 60 60-62. Relations between stream- and velocity-functions. Connection with the theory of complex variables 62 63, 64. Simple types .of motion, acyclic and cyclic. Potential of a row of simple or double sources 66 65, 66. Inverse relations. Examples; confocal curves; flow tcom an open channel 69 67. General formulae; Fourier method 72 68. Motion of a circular cylinder without circulation; stream-lines . . 73 69. Motion of a cylinder with circulation. Trochoidal path under constant force 75 70. Note on more general problems . . . . * . . . . 78 71. Inverse methods. Motion due to the translation of a cylinder; case of an elliptic cylinder. Flow past an oblique lamina; couple-resultant of fluid pressures 78 72. Motion due to a rotating rigid boundary. Rotating prismatic vessel whose section is an ellipse, equilateral triangle, or circular sector. Rotating eUiptic cylinder in infinite liquid. Formula for the most general motion of an elliptic cylinder with circulation .... 82 73. Steady motions with a free surface. Schwarz' method of conformal transformation 86 74. Two-dimensional form of Borda's mouthpiece 88 75. Fluid issuing from a rectilinear aperture. Coefficient of contraction . 90 76. 77. Impact of a stream on a lamina, direct and obUque. Calculation of resistance 92 78. .Bobyleff's problem 96 79. Discontinuous motions 98 80. Flow in a curved stratum 101 X Contents CHAPTER V IRROTATIONAL MOTION OF A LIQUID: PROBLEMS IN THREE DIMENSIONS ART. PAGE 81, 82. Spherical harmonics. Maxwell's theory of poles 103 83. Laplace*8 equation in polar co-ordinates 105 84, 85. Zonal harmonics. Hypergeometric series 106 86. Tesseral and sectorial harmonics 109 87, 88. Conjugate property of surface-harmonics. Expansions . . . Ill 89. Symbolical solutions of Laplace's equation. Definite- integral forms . 112 90, 91. Hydrodynamical applications. Impulsive pressure over a spherical surface. Prescribed normal velocity. Energy of motion generated 114 92, 93. Motion of a sphere in an infinite liquid. Inertia coefficient. Sphere in a liquid with concentric spherical boundary 115 94-96. Stokes* stream-function. Expression in spherical harmonics. Stream- lines of a sphere. Image of a double source in a sphere . 118 97. Bankine's inverse method 122 98, 99. Motion of twq spheres in a liquid; kinematical formulae . . . 123 100, 101. Cylindrical harmonics. Solutions of Laplace's equation in terms of BessePs functions. Expansion of an arbitrary function . . 127 102. Hydrodynamical examples. Flow through a circular aperture. Inertia coefficient of a circular disk 130 103-106. Ellipsoidal harmonics for an ovary ellipsoid. Solutions of Laplace's equation. Applications to the motion of an ovary eUipsoid in liquid 133 107-109. Harmonics for a planetary ellipsoid. Flow through a circular aperture. Stream-lines of a circular disk. Translation and rotation of a plane- tary ellipsoid 137 110. Motion of a fluid in an ellipsoidal envelope 141 111. General expression for V'(^ in orthogonal co-ordinates .... 142 112. Confocal quadrics; ellipsoidal co-ordinates 143 113. Flow through an elliptic aperture 145 114,115. Translation and rotation of an ellipsoid in liquid ; inertia coefficients . 147 116. References to other problems 150 CHAPTEE VI ON THE MOTION OF SOLIDS THROUGH A LIQUID: DYNAMICAL THEORY 117, 118. Kinematical formulae for the case of a single body . . . .151 119. Theory of the 'impulse' 153 120. Dynamical equations relative to axes fixed in the solid . .154 121. Kinetic energy; coefficients of inertia 155 122. 123. Components of impulse. Reciprocal formulae 156 124. The three permanent translations; stability 158 125. The possible types of steady motion. Motion due to an impulsive couple 160 126. Hydrokinetic symmetry .... 162 Contents xi ART. PAGE 127-129. Motion of a solid of revolution. Stability of motion parallel to an axis of symmetry. Influence of rotation. Other types of steady motion 165 130. Motion of a 'heUcoid' 170 131. Inertia-coefficients of a liquid contained in a moving rigid envelope . 171 132-134. Case of a perforated solid with cyclic motion through the apertures. Steady motion of a ring; condition of stability . . . .171 135. Lagrange's equations of motion in generalized co-ordinates. Hamiltonian principle 175 136. Adaptation to hydrodynamics 178 137. 138. Motion of a sphere near a rigid boundary. Motion of two spheres in the line of centres 181 130. Modification of Lagrange's equations in the case of cyclic motion; ignoration of co-ordinates 183 140, 141. Equations of motion of a gyrostatic system 186 142. Kineto-statics 189 143, 144. Motion of a sphere in a cyclic region. Circulation round thin cores or through tubes; comparison with electromal^netic phenomena 191 CHAPTER VII VORTEX MOTION 146. * Vortex-lines' and * vortex-filaments'; kinematical properties . . 194 146. Persistence of vortices; Kelvin's proof. Equations of Cauchy, Stokes, and Helmholtz. Motion in a fixed ellipsoidal vessel, with uniform vorticity 196 147. Conditions of determinateness in vortex motion 200 148. 149. Velocity expressed in terms of expansion and vorticity; electromagnetic analogy. Case of an isolated vortex 201 150. Velocity-potential due to a vortex 204 151. Vortex-sheets 206 152. Impulse of a vortex-system 208 153. Formulae for the kinetic energy 209 154. 155. Rectilinear vortices. Stream-lines of a vortex-pair. Other examples . 213 156. Investigation of the stability of a row of vortices ; and of a double row . 218 157. Creneral formulae relating to a rectilinear vortex-system. Kirchhoff's theory 223 158. Stability of a cylindrical vortex 224 159. Kirchhoff's eUiptic vortex 226 160. Vortices in a curved stratum of fluid 227 161-163. Circular vortices. Potential- and stream-function of an isolated vortex- ring; stream-lines. Impulse and energy; velocity of translation of a vortex-ring 227 164. Mutual influence of vortex-rings. Image of a vortex-ring in a sphere 233 165. General conditions for steady motion of a fluid. Cylindrical and spherical vortices 235 166. References 238 167. Clebsch's transformation of the hydrodynamical equations . . . 239 xii Contents CHAPTER VIII TIDAL WAVES ABT. PAOB 168. General theory of small oscillations; normal modes ; forced oscillations 241 169-174. Free waves in uniform canal ; wave-velocity ; effect of initial conditions ; physical meaning of the various approximations; energy of a wave- system 246 176. Artifice of steady motion 253 176. Superposition of wave-systems; reflection 253 177-179. Effect of disturbing forces ; free and forced oscillations in a finite canal 254 180-184. Canal theory of the tides ; disturbing potential. Tides in an equatorial canal, and in a canal parallel to the equator; semi-diurnal and diurnal tides. Canal coincident with a meridian ; change of mean level; fortnightly tide. Equatorial canal of finite length; lag of the tide 258 185, 186. Waves in a canal of variable section. Examples of free and forced oscillations; exaggeration of tides in shallow seas and estuaries . 265 187, 188. Waves of finite amplitude ; change of type in a progressive wave. Tides of the second order 269 189, 190. Wave-motion in two horizontal dimensions; general equations. Oscil- lations of a rectangular sheet of water 274 191,192. Oscillations of a circular sheet; Bessel's functions ; contour-lines . . 276 193. Case of variable depth. Circular basin 283 194, 195. Propagation of disturbances outwards from a centre; Bessel's function of the second kind. Waves due to a periodic local pressure . 284 196, 197. General formula for diverging waves. Example of a transient local disturbance 290 198-201. Oscillations of a spherical sheet of water ; free and forced waves. Effect of the mutual attraction of the liquid. Reference to the case of a sea bounded by meridians and parallels 294 202, 203. General equations of motion of a d3mamical system relative to rotating axes 300 204. Application to the small oscillations of a rotating system . . . 302 205, 206. Free oscillations ; stability, 'ordinary* and 'secular.' Forced oscillations 303 207, 208. Application to hydrod3mamics ; tidal oscillations of a rotating plane sheet of water; waves in a straight canal 308 209-211. Circular sheet of uniform depth; free and forced oscillations . .311 212. Circular basin of variable depth 316 213, 214. Tidal oscillations on a rotating globe. Equations of Laplace's kinetic theory 318 215-217. Case of symmetry about the axis. Tides of long period . . .321 218-221. Diurnal and semi-diurnal tides. Discussion of Laplace's solution . . 328 222,223. Hough's investigations; extracts and results 335 224. Modifications of the kinetic theory due to the actual configuration of the ocean; questions of phase 341 225, 226. Stability of the ocean. Remarks on the general theory of kinetic stability 343 Appendix: On Tide-generating forces 346 Contents xiii CHAPTER IX SURFACE WAVES ABT. PAOB 227. Statement of the two-dimenflional problem; surface-conditions . . 351 228. Standing waves; lines of motion 352 229, 230. Progressive waves ; orbits of particles. Wave- velocity ; numerical tables. Energy of a simple-harmonic wave-train . . . . 354 231, 232. Artifice of steady motion. Oscillations of superposed liquids . 359 233, 234. Waves on the boundary between two currents ; instability . . 363 235. Waves in a heterogeneous liquid 367 236, 237. Theory of 'group- velocity.' Propagation of energy .... 369 238-240. The Cauchy-Poisson wave-problem; waves due to an initial local elevation, or to a local impulse 373 241. Kelvin's approximate formula for the effect of a local disturbance in a medium; graphical constructions 384 242-246. Surface-disturbance of a stream. Case of finite depth. Effect of inequalities in the bed 388 247. Waves due to a submerged cylinder 401 248. General theory of waves due to a travelling disturbance; approximate formidae 403 249. Wave-resistance. Example of the submerged cylinder . . . 407 250. Surface-waves of finite height. Stokes' waves of permanent type. Limiting form 409 251. Gerstner's rotational waves 412 252. 253. Solitary waves. Oscillatory waves of Korteweg and De Vries . . 415 254. Hebnholtz' dynamical condition for waves of permanent type . . 420 255. Wave-propagation in two horizontal dimensions. Effect of a localized initial disturbance 422 256. Effect of a pressure-disturbance advancing over the surface of water; wave-pattern; ship- waves 426 257. Standing waves in limited masses of water. Case of uniform depth. Oscillations in a rectangular tank with a cylindrical obstacle . 432 258. 259. Transverse oscillations in channels of triangular and semicircular section 435 260,261. Longitudinal oscillations; cases of triangular section. Edge- waves . 438 262-264. Oscillations of a liquid globe; lines of motion. Ocean of uniform depth on a spherical nucleus 443 265. Capillarity; surface-condition 448 266. Capillary waves; group- velocity 449 267. Waves under both gravity and capillarity; minimum wave- velocity . 452 268. Waves on the common boundary of two currents .... 455 269. Waves due to a local disturbance. Effect of a travelling disturbance; waves and ripples 456 270,271. Surface-disturbance of a stream; formal investigation . . 457 272. Effect of a pressure-point advancing over the surface of water; fish-line problem; wave-patterns 462 273, 274. Vibrations of a cylindrical column of liquid; instability of a jet . 465 275. Oscillations of a liquid globule, and of a bubble 468 xiv Contents CHAPTER X WAVES OP EXPANSION ABT. PAOB 276-280. Plane waves; velocity of sound; energy of a wave-system . . 470 281-284. Plane waves of finite amplitude; methods of Biemann and Eamshaw. Condition for permanence of type ; Bankine's investigation. Question as to the possibility of a wave of discontinuity .... 475 285, 286. Spherical waves. Solution in terms of initial conditions . . . 483 287, 288. General equation of sound-waves. Poisson's integral Equation of energy. Determinateness of solutions 486 289. Simple-harmonic vibrations. Sources of sound, simple and double; emission of energy 489 290. Helmholtz' adaptation of Green' s theorem. Velocity-potential in terms of surface-distributions of sources. Kirchhoff*s formula . . .491 291. Periodic disturbing forces 495 292. Applications of spherical harmonics; general formulae . . . 497 293. Vibrations of air in a spherical envelope. Vibrations of a spherical stratum 500 294. Propagation of waves outwards from a spherical surface ; effect of lateral motion 502 295. Theory of the ball-pendulum; correction for inertia; coefficient of dissipation (due to air-waves) 504 296-298. Scattering of sound-waves by a sphere. Impact of waves on a moveable sphere; case of s3mchronism 505 299, 300. Approximate treatment of diffraction problems when the wave-length is relatively large. Diffraction by a flat disk, by an aperture in a thin screen, and by a small obstacle of any form 510 301. Solution of the equation ^ =c^v^<f> in spherical harmonics. Conditions at the front of a diverging wave 517 302. Sound-waves in two dimensions ; effect of a transient source ; comparison with the one- and three-dimensional cases 519 303. Simple-harmonic vibrations ; solutions in Bessel's functions. Oscillating cylinder 523 304. Scattering of waves by a cylindrical obstacle 525 305. Approximate theory of diffraction of long waves in two dimensions. Diffraction by a flat blade, and by an aperture in a thin screen . 527 306. 307. Reflection and transmission of sound-waves by a grating . . . 529 308. Diffraction by a semi-infinite screen 535 309, 310. Waves propagated vertically in the atmosphere; 'isothermal' and *convective' hypotheses 538 311. General equations of atmospheric waves 544 312, 313. Two-dimensional case; gravitational oscillations 546 314-316. Large-scale oscillations of an atmosphere surrounding a globe, without. and with rotation. Atmospheric tides 551 Contents xv CHAPTER XI VISCOSITY ABT. PAGE 317y 318. Theory of dissipative forces. One degree of freedom; free and forced oscillations. Effect of friction on phase 556 319. Application to tides in equatorial canal; tidal lag, and tidal friction . 559 320. Equations of dissipative systems in general; frictional and gyrostatic terms. Dissipation function 563 321. Oscillations of a dissipative system about a configuration of absolute equilibrium 564 322. Effect of gyrostatic terms. Example of two degrees of freedom; dis- turbing force of long period 565 323-325. Viscosity of fluids ; specification of stress; formulae of transformation . 567 326, 327. The stresses assumed to be linear functions of rates of strain. Coefficient of viscosity. Boundary conditions; question of slipping . . 569 328. Dynamical equations. The modified Helmholtz equations; interpretation 572 329. Dissipation of energy in a liquid by viscosity 574 330. Problems of steady motion: flow of a liquid between parallel planes; Hele Shaw's experiments 576 331. 332. flow through a pipe of circular section; Poiseuille's laws; question of slipping. Other forms of section 577 333, 334. Steady rotation of a cylinder, and of a sphere. Practical limitation to the solutions 580 335, 336. General solution of the problem of slow steady motion in spherical harmonics. Formulae for the stresses 583 337. Rectilinear motion of a sphere; resistance; terminal ^velocity ; stream- lines. Motion of a liquid sphere. Motion of a solid sphere, with slipping 586 338. Method of Stokes; solutions in terms of the stream-function . .591 339. Steady motion of an ellipsoid 503 340. Questions as to the validity of preceding solutions. Oseen's criticism. Modified investigation 595 341. 342. Steady motion of a liquid in a given field of force. Analogy with the theory of flexure of elastic plates 601 343. Steady motion of a cylinder, treated by Oseen's method. Resistance . 604 344. Dissipation of energy in steady motion; theorems of Helmholtz and Korteweg. Rayleigh's extension 606 345-347. Problems of periodic motion. Laminar motion; diffusion of vorticity. Oscillating plane. Periodic tidal force ; feeble influence of viscosity on rapid motions 609 348, 349. Effect of viscosity on water-waves 614 350, 351. Effect of surface-forces ; generation and maintenance of waves by wind. Scott Russell's observations. Calming effect of oil . . . 619 352, 353. Periodic motion with a spherical boundary ; general solution in spherical harmonics 622 354. Applications; decay of motion in a spherical vessel; torsional oscilla- tions of a hollow sphere containing liquid 628 ^^^^tmmm^m^^am^^^rmmma^mimim^mmmmmmu^tmmmmmf'a^''Bmmmr^!^ u jl xvi Contents ABT. PAGE 356. Osoillations of a liquid globe 630 356. Effect of viscoeity on the rotational oscillations of a sphere, and on the pendulum 632 357. Notes on two-dimensional problems 635 358. Viscosity in gases; dissipation function 636 359. Damping of plane waves of sound by viscosity 637 360. Combined influence of viscosity and thermal conduction . . 639 36L Effect of viscosity on diverging waves 641 362, 363. Effect on the scattering of sound-waves by a spherical obstacle, fixed or free 645 364. Damping of sound-waves in a spherical vessel 649 365, 366. Turbulent motion. Be3mold's experiments ; critical velocities of water in a pipe; law of resistance. Inferences from theory of dimensions 651 367, 368. References to theoretical investigations of Bayleigh and Kelvin . . 655 369. Statistical method of Be3molds 660 370-372. Resistance of fluids. References to various theories. 'Peripteroid' motion. Dimensional formulae 664 CHAPTER XII ROTATING MASSES OF LIQUID 373. Formulae relating to attraction of ellipsoids. Potential energy of an ellipsoidal mass 669 374. Maclaurin's ellipsoids. Relations between eccentricity, angular velocity and angular momentum; numerical tables 671 375. Jacobi's ellipsoids. Linear series of ellipsoidal forms of equilibrium. Numerical results 673 376. Other special forms of relative equilibrium. Rotating annulus . . 677 377. General problem of relative equilibrium; Poincar6's investigation. Linear series of equilibrium forms; limiting forms and forms of bifurcation. Exchange of stabilities 680 378-380. Application to a rotating system. Secular stability of Maclaurin's and Jacobi's ellipsoids. The pear-shaped figure of equilibrium . . 683 381. Small oscillations of a rotating ellipsoidal mass; Poincar6*s method. References 687 382. Dirichlet's investigations; references. Finite gravitational oscillations of a liquid ellipsoid without rotation. Oscillations of a rotating ellipsoid of revolution 689 383. Dedekind^s ellipsoid. The irrotational ellipsoid. Rotating elliptic cylinder 692 384. Free and forced oscillations of a rotating ellipsoidal shell containing liquid. Precession 694 385. Precession of a liquid ellipsoid 698 List of Authobs cited 702 Index 705 H YDEOD YN AMIC S CHAPTER I THE EQUATIONS OF MOTION 1. The following investigations proceed on the assumption that the matter with which we deal may be treated as practically continuous and homogeneous in structure ; i.e. we assume that the properties of the smallest portions into which we can conceive it to be divided are the same as those of the substance in bulk. The fundamental property of a fluid is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface. This property is the basis of Hydrostatics, and is verified by the complete agreement of the deductions of that science with experiment. Very slight observation is enough, however, to convince us that obUque stresses may exist in fluids in motion. Let us suppose for instance that a vessel in the form of a circular cyUnder, containing water (or other liquid), is made to rotate about its aids, which is vertical. If the angular velocity of the vessel be constant, the fluid is soon found to be rotat- ing with the vessel as one soUd body. If the vessel be now brought to rest, the motion of the fluid continues for some time, but gradually subsides, and at length ceases altogether ; and it is found that during this process the portions of fluid which are further from the aids lag behind those which are nearer, and have their motion more rapidly checked. These phenomena point to the existence of mutual actions between contiguous elements which are partly tangential to the common surface. For if the mutual action were everywhere wholly normal, it is obvious that the moment of momentum, about the axis of the vessel, of any portion of fluid bounded by a surface of revolution about this axis, would be constant. We infer, moreover, that these tangential stresses are not called into play so long as the fluid moves as a solid body, but only whilst a change of shape of some portion of the mass is going on, and that their tendency is to oppose this change of shape. L. H. 1 The Equations of Motion [chap. I 2. It is usual, however, in the first instance to neglect the tangential stresses altogether. Their effect is in many practical cases small, and inde- pendently of this, it is convenient to divide the not inconsiderable difficulties of our subject by investigating first the effects of purely normal stress. The further consideration of the laws of tangential stress is accordingly deferred till Chapter xi. If the stress exerted across any small plane area situate at a point P of the fluid be wholly normal, its intensity (per unit area) is the same for all aspects of the plane. The foUowing proof of this theorem is given here for purposes of reference. Through P draw three straight lines PA, PB, PC mutually at right angles, and let a plane whose direction-cosines relatively to these lines are {, m, n, passing infinitely close to P, meet them in A, By C. Let p, pi, p^f p^ denote the intensities of the stresses* across the faces ABC, PBC, PC A, PAB, respectively, of the tetrahedron PABC, If A be the area of the first-mentioned face, the areas of the others are, in order, ?A, mA, nA. Hence if we form the equation of motion of the tetrahedron parallel to PA we have p^.lA^ pi . A, where we have omitted the terms which express the rate of change of momentum, and the component of the extraneous forces, because they are ultimately propor- tional to the mass of the tetrahedron, and therefore of the third order of small linear quantities, whilst the terms retained are of the second. We have then, ultimately, p== Pi, and similarly p = p^ = p^^ which proves the theorem. 3. The equations of motion of a fluid have been obtained in two different forms, corresponding to the two ways in which the problem of determining the motion of a fluid mass, acted on by given forces and subject to given conditions, may be viewed. We may either regard as the object of our investigations a knowledge of the velocity, the pressure, and the density, at all points of space occupied by the fluid, for all instants ; or we may seek to determine the history of every particle. The equations ^obtained on these two plans are conveniently designated, as by German mathematicians, the 'Eulerian' and the 'Lagrangian' forms of the hydrokinetic equations, although both forms are in reality due to Eulerf. * ReokoDed positiTe when preesures, negatiTe when tenaioDB. Most fluids are, however, incapable under otdinaiy conditions of supporting more than an exceedingly slight degree of tension, so that p is nearly always positive. t "Principes g6n6ranx du mouvement des floides," Hist, de VAcad, de Berlin, 1755. "De principxis motus fluidorum," Nwi Comm, Aead, Pelrap. t. xiv. p. 1 (1759). Lagrange gave three investigations of the equations of motion; first, incidentally, in 2-5] Btderian Equations 8 The Eulerian Equations. 4. Let u, V, w be the components, parallel to the co-ordinate axes, of the velocity at the point (x, y^ z) at the time t. These quantities are then functions of the independent variables Xy y, z, t. For any particular value of t they e^^press the motion at that instant at all points of space occupied by the fluid; whilst for particular values of x, y, z they give the history of what goes on at a particular place.' We shall suppose, for the most part, not only that ti, v, w are finite and continuous functions of x, y, z, but that their space-derivatives of the first order {du/dx, dv/dx, dw/dx, &c.) are everywhere finite*; we shall understand by the term 'continuous motion,' a motion subject to these restrictions. Cases of exception, if they present themselves, will require separate examina- tion. In continuous motion, as thus defined, the relative velocity of any two neighbouring particles P, P' wiU always be infinitely smaU, so that the line PP' will always remain of the same order of magnitude. It follows that if we imagine a small closed surface to be drawn, surrounding P, and suppose it to move with the fluid, it will always enclose the same matter. And any surface whatever, which moves with the fluid, completely and permanently separates the matter on the two sides of it. 5. The values of u, v, w for successive values of t give as it were a series of pictures of consecutive stages of the motion, in which however there is no immediate means of tracing the identity of any one particle. To calculate the rate at which any function F (x, y, z, t) varies for a moving particle, we remark that at the time t + St the particle which was originally in the position {x, y, z) is in the position {x + 1^8^ y + vht^ z + wU), so that the corresponding value of P is 7^W f^JP rUP rUP P (a: -t- w8<, y 4- v8t, z + wU, <4.&)«P-fw8i — -hi;&— +w;&|i4.8<^. If, after Stokes, we introduce the symbol D/Dt to denote a differentiation foUowing the motion of the fluid, the new value of P is also expressed by P + DF/Dt . 8t, whence DF ap. ap. ap. ap :D^ = -a7+^ax+"a^ + ^a^ (^) connection with the princijde of Least Action, in the MisceUanea Taurinenata, t. ii. (1760) [Oeuvres, Paris, 1807-92, t. i.]; secondly in his "M^moife snr la Theorie du Honvement dee Fioides," Nouv. ttUm. de VAcad. de Berlin^ 1781 [Otuvres, t. iv.]; and thirdly in the Micaniq%e Anahftique. In this last exposition he starts with the second form of the equations (Art. 14, below), but translates them at once into the * Eulerian ' notation. * It is important to bear in mind, with a view to some later developments mider the head of Vortex Motion, that these derivatiTes need not be assumed to be continuous. 1—2 The Equations of Motion [chap. I 6. To form the dynamical equations, let p be the pressure, p the density, Xy y, Z the components of the extraneous forces per unit mass, at the point (x, y, z) at the time L Let us take an element having its centre at {Xy y, z), and its edges 8x, By, Sz parallel to the rectangular co-ordinate axes. The rate at which the rc-component of the momentum of this element is increasing is p8xBy8zDu/Dt; and this must be equal to the a^-component of the forces acting on the element. Of these the extraneous forces give pSxSySzX. The pressure on the yz-iace which is nearest the origin will be ultimately that on the opposite face {P + h^Pl^^ • Src) Syhz. The difference of these gives a resultant — dp/dx , SxSySz in the direction of x-positive. The pressures on the remaining faces are perpendicular to x. We have then phx8yiz-jr- = pSxSySzX — ^8xSy8z. Substituting the value of Du/Dt from (1), and writing down the symmetrical equations, we have du du du du du ^ 1 dp pdx' dv dv dv dv ^, 1 dp dx dv dx dy dv dy dz dv Wz pdy' y dw , dw , dw , dw „ ^dp (2) 7. To these dynamical equations we must join, in the first place, a certain kinematical relation between w, v, w, p, obtained as follows. If V be the volume of a moving element, we have, on account of the constancy of mass,*/ D.pv Dt = 0, or 1^+1^ = (1) To calculate the value of 1/v . D^/Dt, let the element in question be that which at time t fills the rectangular space SxSySz having one corner P at (x, y, z), and the edges PL, PM, PN (say) parallel to the co-ordinate axes. At time t -\- Bt the same element will form an oblique parallelepiped, and since * It is easily seen, by Taylor's theorem, that the mean pressure over any face of the element dxSy&t may be taken to be eqnal to the pressure at the centre of that face. ^ ... / V0<« 6-7J Uqtiation of Continuity 5 the velocities of the particle L relative to the particle P are dujdx . Sx, dvjdx . Sx, dwjdx . 8x, the projections of the edge PL become, after the time 8f, respectively. To the first order in &, the length of this edge is now (l+|«)8.. and similarly for the remaining edges. Since the angles of the parallelepiped differ infinitely little from right angles, the volume is still given, to the first order in 8i, by the product of the three edges, i.e. we have \D^ du dv dw ,n\ °' ^Dt'di^Ty^Tz (^^ Hence (1) becomes Dp (du dv dw\ .ox This is called the * equation of continuity.' The expression g^ + ^+a^ W which, as we have seen, measures the rate of increase of volume of the fluid at the point (a?, y, 2), is conveniently called the ^expansion' at that point. From a more general point of view the expression (4) is called the * divergence' of the vector (u, v, w); it is often denoted briefly by div (w, V, w). The preceding investigation is substantially that given by Euler*. Another, and now more usual, method of obtaining the equation of continuity is, instead of following the motion of a fluid element, to fix the attention on an element 8x8ySz of space, and to calculate the change produced in the included mass by the flux across the boundary. If the centre' of the element be at {x, y, z), the amount of matter which per unit time enters it across the y2;-face nearest the origin is and the amount which leaves it by the opposite face is (■ />w + i - g|^ Bxj 8y8z. * 2.C. ante p. 2. 6 The Equations of Motion [chap, i The two faces together give a gain per unit time. Calculating in the same way the effect of the flux across the remaining faces, we have for the total gain of mass, per unit time, in the space hxhyhz, the formula Since the quantity of matter in any region can vary only in consequence of the flux across the boimdary, this must be equal to -^^{phxhyhz), whence we get the equation of continuity in the form a< "*" ~a^ + " ajT "^ "ar - *^ ^^^ 8. It remains to put in evidence the physical properties of the fluid, so far as these affect the quantities which occur in our equations. In an 'incompressible' fluid, or liquid, we have Dp/Dt == 0, in which case the equation of continuity takes the simple form s^-i-i-"- <■' It is not assumed here that the fluid is of uniform density, though this is of course by far the most important case. If we wished to take account of the slight compressibility of actual liquids, we should have a relation of the form p = K{p- po)/po. • • • (2) or pIPo^I + P/k, (3) where k denotes what is called the 'elasticity of volume.' In the case of a gas whose temperature is uniform and constant we have the 'isothermal' relation pIpo^pIpo W where p^, pQ are any pair of corresponding values for the temperature in question. In most cases of motion of gases, however, the temperature is not constant, but rises and falls, for each element, as the gas is compressed or rarefied. When the changes are so rapid that we can ignore the gain or loss of heat by an element due to conduction and radiation, we have the 'adiabatic' relation pIPo = (p/Po)^y (5) 7-9] Boundary Condition 7 where Pq and p^ are any pair of corresponding values for the element con- sidered. The constant y is the ratio of the two specific heats of the gas ; for atmospheric air, and some other gases, its value is 1*408. 9. At the boundaries (if any) of the fluid, the equation of continuity is replaced by a special surface-condition. Thus at a fixed boundary, the velocity of the fluid perpendicular to the surface must be zero, i.e. if I, m, n be the direction-cosines of the normal, lu-i- mv -{- nw = (1) Again at a surface of discontinuity, i.e. a surface at which the values of Uf V, w change abruptly as we pass from one side to the other, we must have I (wi — w,) -f- m (vi — Vj) + n (t<^i — w;^) = 0, (2) where the suffixes are used to distinguish the values on the two sides. The same relation must hold at the common surface of a fluid and a moving soUd. The general surface-condition, of which these are particular cases, is that if F {x^ y, z, ^) = be the equation of a boimding surface, we must have at every point of it DFIDt = Q (3) For the velocity relative to the surface of a particle lying in it must be wholly tangential (or zero), otherwise we should have a finite flow of fluid across it. It follows that the instantaneous rate of variation of F for a surface-particle must be zero. A fuller proof, given by Lord Kelvin*, is as follows. To find the rate of motion (v) of the surface F (x, y, 2, t) = 0, normal to itself, we write F(x-\- IvU, y + mvU, z + nvht, « + SO = 0, where Z, m, n are the direction-cosines of the normal at (x, y, 2), whence .( ,dF ^ dF^ dF\dF - Since (I,m.«) = (^. ^. -g- j - B, we have ^ '^ "" »"Sf" ^^^ At every point of the surface we must have v = lu + mv -\- nw, which leads, on substitution of the above values of {, m, n, to the equation (3). * (W. Thomaon) "Notes on Hydrodynamios,*' Comb, and Dub. Ifaih. Joum. Feb. 1848. [MaOiemiUical and Physical Papers, Cambridge, 1882..., t. i. p. 83.] 8 The EqtMtions of Motion [chap, i The partial differential equation (3) is also satisfied by any surface moving with the fluid. This follows at once from the meaning of the operator DjDt, A question arises as to whether the converse necessarily holds; i.e. whether a moving surface whose equation -F = satisfies (3) will always consist of the same particles. Considering any such surface, let us fix our attention on a particle P situate on it at time t. The equation (3) expresses that the rate at which P is separating from the surface is at this instant zero ; and it is easily seen that if the motion be oontinv/ous (according to the definition of Art. 4), the normal velocity, relative to the moving surface -F, of a particle at an infinitesimal distance t, from it is of the order ^, viz. it is equal to Gt, where G is finite. Hence the equation of motion of the particle P relative to' the surface mav be written Dtim = Gi This shews that log t, increases at a finite rate, and since it is negative infinite to begin with (when t, = 0), it remains so throughout, i.e. f remains zero for the particle P. The same result follows from the nature of the solution of dF dF dF dF ^ considered as a partial differential equation in ^*. The subsidiary system of ordinary differential equations is dt=^=^=^ (6) in which x, y, z are regarded as functions of the independent variable t. These are evidently the equations to find the paths of the particles, and their integrals may be supposed put in the forms x=fi (a, b, c, t), y=f^ (a, 6, c, t), z=U(a,h,c,t\ (7) where the arbitrary constants a, 6, c are any three quantities serving to identify a particle ; for instance they may be the initial co-ordinates. The general solution of (5) is then found by elimination of a, h, c between (7) and JP = Vr (a, 6, c) (8) where ^ is an arbitrary function. This shews that a particle once. in the surface F=0 remains in it throughout the motion. Eqiuxtion of Energy, 10. In most cases which we shall have occasion to consider the extraneous forces have a potential ; viz. we have an an an * Lagrange, Oeuvres, t. iv. p. 706. 9-10] Energy 9 The physical meaning of fl is that it denotes the potential energy, per unit mass, at the point (2;, y, z), in respect of forces acting at a distance. It will be sufficient for the present to consider the case where the field of extraneous force is constant with respect to the time, i,e. dO./dt = 0. If we now multiply the equations (2) of Art. 6 by u, v, w, in order, and add, we obtain a result which may be written If we multiply this by hxhyhz^ and integrate over any region, we find |(r+F).-///(.| + .| + »|)^*^ (2, where T = J///p (w« + i;« -{- 1(;«) (fodydz, V = Siiilpdzdydz, ...(3) i.e. T and V denote the kinetic energy, and the potential energy in relation to the field of extraneous force, of the fluid which at the moment occupies the region in question. The triple integral on the right-hand side of (2) may be transformed by a process which will often recur in our subject. Thus, by a partial integration, jjj af ^^^y^^ ^ jj^^ ^y^^ "■ jjjP gix ^^^y^^> where [pu\ is used to indicate that the values of jni at the points where the boimdary of the region is met by a line parallel to x are to be taken, with proper signs. If Z, m, n be the direction-cosines of the inwardly directed normal to any element SjS of this boundary, we have hyhz — ± 188, the signs alternating at the successive intersections referred to. We thus find that !S[jm]dydz = - J/pw IdS, where the integration extends over the whole boimding surface. Trans- forming the remaining terms in a similar manner, we obtain ^{T + V)=jfp{lu + mv + nw)dS + jjjp(^+^ + ^')dxdydz...{4:) In the case of an incompressible fluid this reduces to the form ^1 S^^ "*" ^^ " f/^'^ + mv 4- nw)pdS (5) Since Zw -f- mt? -h nw denotes the velocity of a fluid particle in the direction of the normal, the latter integral expresses the rate at which the pressures fhS exerted from without on the various elements hS of the boundary are doing work. Hence the total increase of energy, kinetic and potential, of any 10 The Equations of Motion [chap, i portion of the liquid, is equal to the work done by the pressures on its surface. In particular, if the fluid be bounded on all sides by fixed waUs, we have over the boundary, and therefore T 4- F = const (6) A similar interpretation can be given to the more general equation (4), provided p be a function of p only. If we write --Hi) i'» E then E measures the work done by unit mass of the fluid against external pressure, as it passes, under the supposed relation between p and p, from its actual volume to some standard volume. For example, if the unit mass were enclosed in a cylinder with a sliding piston of area A, then when the piston is pushed outwards through a space 8x, the work done is pA . Sx, of which the factor ASx denotes the increment of volume, i.e, of p"^. In the case of the adiabatic relation we find ^ = -1^(2-20) (8) We may call E the intrinsic energy of the fluid, per unit mass. Now, recalling the interpretation of the expression du dv dw dx dy dz given in Art. 7, we see that the volimie-integral in (4) measures the rate at which the various elements of the fluid are losing intrinsic energy by expansion*; it is therefore equal to — DW/Dt, where W = /// Epdxdydz (9) Hence (^t -{- V + W) ^jjp {lu 4- mv + nw)dS (10) • The total energy, which is now partly kinetic, partly potential in relation to a constant field of force, and partly intrinsic, is therefore increasing at a rate equal to that at which work is being done on the boundary by pressure from without. 10-11] ImpuUive Motion 11 Impulsive Generation of Motion. 11. If at any instant impulsive forces act bodily on the fluid, or if the boundary conditions suddenly change, a sudden alteration in the motion may take place. The latter case may arise, for instance, when a soUd immersed in the fluid is suddenly set in motion. Let p be the density, u, t;, w the component velocities immediately before, u', v\ w' those immediately after the impulse, X\ Y\ Z' the components of the extraneous impulsive forces per unit mass, m the impulsive pressure, at the point (x, y, z). The change of momentum parallel to x of the element defined in Art. 6 is then pSxSySz i^' — vl) ; the x-component of the extraneous impulsive forces is p8:s8ySz X\ and the resultant impulsive pressure in the same direction is — 'dmfdx . hxhyhz. Since an impulse is to be regarded as an infinitely great force acting for an infinitely short time (r, say), the effects of all finite forces during this interval are neglected. Hence, pSxSySz {u' — w) = pSxSySz-X' — -^ &r%8z, or Similarly, , «., 13tD u — w=A — ^— . pox pdy W — W = L -^- , p oz > (1) These equations might also have been deduced from (2) of Art. 6, by multiplying the latter by 8^, integrating between the limits and r, putting X'^Txdt, Y'^Trdt, Z'^Tzdt, w^Wdt, JO Jo Jo Jo and then ma]dng r vanish. In a Uquid an instantaneous change of motion can be produced by the action of impulsive pressures only, even when no impulsive forces act bodily on the mass. In this case we have X\ Y\ Z' = 0, so that , 13tD u — w = — ^5— p ox , 13oj . pdy ^ , I dm p oz \ (2) If we differentiate these equations with respect to x, y, 2;, respectively, and 12 The Bqtiations of Motion [chap, i add, and if we further suppose the density to be uniform, we find by Art. 8(1) that d^w d^w d^rn 4- 4- = The problem then, in any given case, is to determine a value of m satisfying this equation and the proper boundary conditions* ; the instantaneous change of motion is then given by (2). Equations referred to Moving Axes. 12. It is sometimes convenient in special problems to employ a system of rectangular axes which is itself in motion. The motion of this frame may be specified by the component velocities n, v, w of the origin, and the com- ponent rotations p, q, r, all referred to the instantaneous positions of the axes. If u, V, w be the component velocities of a fluid particle at (x^ y, z), the rates of change of its co-ordinates relative to the moving frame will be ^ = u-u + Ty-qz, -^ = t;-v + pz-rx, ^^w-w + qx-i^y, . ,(l) After a time ht the velocities of the particle parallel to the new positions of the co-ordinate axes will have become To find the component accelerations we must resolve these parallel to the original positions of the axes in the manner explained in books on Dynamics. In this way we obtain the expressions du duDx duDy duDz ' dv dv Dx , dv Dy , dv Dz \ /«x dt-^+'^+diDt + TylA + diDf^ (^) dw dwDx dtoDy dwDz _ ^Qw + pv +__ + — _+ ^ — .^ These will replace the expressions in the left-hand members of Art. 6 (2)'|'. The general equation of continuity is hU'^Ki^^hU^^h" <*) * It will appear in Chapter m. that the value of w is thus detemuDate, save as to an additive constant. t GreenhiU, "On the General Motion of a Liquid EUipsoid. . .," Proc, Camb. PhU. Soe. t. iv. p. 4 (1880). 11-14] Lagrangian Eqvxztions 13 reducing in the case of incompressibility to the form du dv dw di^d-y^Tz^^ (^) as before. The Lagrangian Equations. 13. Let a, 6, c be the initial co-ordinates of any particle of fluid, x, y, z its co-ordinates at time L We here consider x, y, z bls functions of the independent variables a, b^ Cyt; their values in terms of these quantities give the whole history of every particle of the fluid. The velocities parallel to the axes of co-ordinates of the particle (a, b, c) at time t are dx/dt, dy/dt, dz/dt, and the component accelerations in the same directions are dh>/dt\ dhf/dt\ dh/dfi. Let p be the pressure and p the density in the neighbourhood of this particle at time t; X, Y, Z the components of the extraneous forces per imit mass acting there. Considering the motion of the mass of fluid which at time t occupies the differential element of volume Sxiyhz, we find, by the same reasoning as in Art. 6, X- Idp pdx' Y- Idp Z- Idp pdz' These equations contain differential coefficients with respect to x, y, z, whereas onr independent variables are a, b, c, t. To eliminate these dif- ferential coefficients, we multiply the above equations by dxjda, dy/da, dzjda, respectively, and add ; a second time by dxjdb, dy/db, dz/db, and add ; and again a third time by dxfdc, dyjdc, dz/dc, and add. We thus get the three equations \dt* "^Jda^Kdt* ^)da'^\dt* ^Jda'^pda'^' \dt* "^Jdb^Kdt* )db^\dt*~^)db^ pdb~^' 'd*x_ \dx (dhi y\3.y,/a»e \dz idp_ \di* ^)d'c'^\dr*~^)dE^W-^)d'c'^~pd'c-^- These are the 'Lagrangian' forms of the dynamical equations. 14. To find the form which the equation of continuity assumes in terms of our present variables, we consider the element of fluid which originally occupied a rectangular parallelepiped having its centre at the point (a, 6, c), and its edges 8a, S6, 8c parallel to the axes. At the time t the same element forms an oblique parallelepiped. The centre now has for its co-ordinates 14 The Equations of Motion [chap. I X, y, z; and the projections of the edges on the co-ordinate axes are respectively 3^^' i^'*' d^^' 36^*' 06^' db^^' dx ^, dy ^ 3z a ;^~ OC» ^ OCm x~ oc» oc oc oc The volume of the parallelepiped is therefore dx dy dz 3a ' da' da dx dy db' 36' dx dy dc* dj_ db dz^ dc 8ahbSc, or, as it is often written, (a, 6, c) Hence, since the mass of the element is unchanged, we have ^ 3 (.r, y, z ) ^dWhTcr^'^ where />o is the initial density at (a, 6, c). In the case of an incompressible fluid p = Po, ^o that (1) becomes 3 (x, y, z) (1) 3 (a, 6, c) = 1. (2) Weber's Transformation. 15. If as in Art. 10 the forces X, Y, Z have a potential H, the dynamical equations of Art. 13 may be written d^dx d*ydy dhdz 3a_13p . , dt^ da ^ dt^ da ^ dt^ da~ da pda' *^" *^- Let us integrate these equations with respect to t between the limits and L We remark that I f*d^dx^^ r3x3x"]<_ f^dx Jodt^da "[dtdajo Jodt d*x dt ' __dxdx - 3 [^ fdx\* ^ "dtd^^'^'^^^diJoKdi) ""' 14-16] Weber's Transformation 15 where t^o is the initial value of the x-component of velocity of the particle (a, 6, c). Hence if we write we find* -/:[/?^"-M(S)'^(IA©T dx dx dy dy dzdz _ dx. *. (1) ^ax ayay^^_ __ax.i /ov a<a6'^a<a6'*'a<a6 *"~ a6''" ^^ dxdx dydy dzdz _ a^ a< ac "^ ^ac ■*■ a< ac ~ "'» ~ ~ ac • / These three equations, together with i-/?-"-*{(i)'-(i)'-(ir} (') and the equation of continuity, are the partial differential equations to be satiBfied by the five unknown quantities x, y, z, p, x'y P being supposed already eliminated by means of one of the relations of Art. 8. The initial conditions to be satisfied are 16. It is to be remarked that the quantities a, 6, c need not be restricted to mean the initial co-ordinates of a particle ; they may be any three quanti- ties which serve to identify a particle, and which vary continuously from one particle to another. If we thus generaUze the meanings of a, 6, c, the form of the dynamical equations of Art. 13 is not altered ; to find the form which the equation of continuity assumes, let Xq, y^, z^ now denote the initial co-ordinates of the particle to which a, 6, c refer. The initial volume of the parallelepiped, whose centre is at (xq, y^, Zq) and whose edges correspond to variations 8a, 86, 8c of the parameters, a, 6, c, is a (a, 6, c) so that we nave p ^. — j^ — r = po -5-7 — . x , (1) "^ d (a, 6, c) '^" d (a, 6, c) ' y or, for an incompressible fluid, 8 (a?> y, g) ^ 3 (a? o>yo, gp) ^2) 9 (a, 6, c) 3 (a, 6, c) ^ ' * H. Weber, **Ueber eine Traoaformation der hydiodyiianuBoheii Gleichungen," Crelle, t. Ixviii (1868). It is assumed in (1) that the density p, if not uniform, is a function of p only. CHAPTER II INTEGRATION OF THE EQUATIONS IN SPECIAL CASES 17. In a large and important class of cases the component velocities tiy V, w can be expressed in terms of a single function <f>, as follows : Such a function is called a 'velocity-potential/ from its analogy with the potential function which occurs in the theories of Attractions, Electro- statics, &c. The general theory of the velocity-potential is reserved for the next chapter; but we give at once a proof of thie following important theorem : If a velocity-potential exist, at any one instant, for any finite portion of a perfect fluid in motion under the action of forces which have a potential, then, provided the density of the fluid be either constant or a function of the pressure only, a velocity-potential exists for the same portion of the fluid at all instants before or after f. In the equations of Art. 15, let the instant at which the velocity- potential <f>Q exists be taken as the origin of time; we have then ^ 4 throughout the portion of the mass in question. Multiplying the equa- tions (2) of Art. 15 in order by (Ja, 3h^ dc, and adding, we get ^ (fo -f P^ rfy + gy rf* - {u^da 4- v^db -f t^o*^) = — dx, s * The reasons for the introduction of the mtniM sign are stated in the Preface. t Lagrange, "M^moire sur la Thtorie dii Mouvement des Eluides," Nouv. mim. de VAcad, de Berlin, 1781 [Oeuvrea, t. iv. p. 714]. The argument is reproduced in the Micanique Analytique, Lagrange's statement and proof were alike imperfect; the first rigorous demonstration is due to Caucby, **M6moire sur la Throne des Ondes," M^. de VAcad. roy. des Sciences, t. i. (1827) [Oeuvres Computes, Paris, 1882 ..,,V* S^rie, t. i. p. 38] ; the date of the memoir is 1815. Another proof is given by Stokes, Cawb. Trans, t. viii. (1845) (see also Math, and Pkys. Papers, Cam- bridge, 1880. . ., t. i pp. 106, 158, and t. ii. p. 36), together with an exceUent historical and critical account of the whole matter. 17^18] Velocity-Potential 17 or, in tha'Eulerian' notation, udx + vdy + wdz = — <? (^o + x) ~ "~ ^» ^^7' Since the upper limit of t in Art. 15 (1) may be positive or negative, this proves the theorem. It is to be particularly noticed that this continued existence of a velocity- potential is predicated, not of regions of space, but of portions of matter. A portion of matter for which a velocity-potential exists moves about and carries this property with it, but the part of space which it originally occupied may, in the course of time, come to be occupied by matter which did not originally possess the property, and which therefore cannot have acquired it. The class of cases in which a velocity-potential exists includes all those where the motion has originated from rest under the action of forces of the kind here supposed ; for then we have, initially, u^da + v^db + '^Jb^dc = 0, or ff>Q = const. The restrictions under which the above theorem has been proved must be carefully remembered. It is assumed not only that the extraneous forces X, y , Z, estimated at per unit mass, have a potential, but that the density p is either uniform or a function of p only. The latter condition is violated, for example, in the case of the convection currents generated by the unequal appUcation of heat to a fluid; and again, in the wave-motion of a hetero- geneous but incompressible fluid arranged originally in horizontal layers of equal density. Another case of exception is that of 'electro-magnetic rotations'; see Art. 29. 18. A comparison of the formulae (1) with the equations (2) of Art. 11 leads to a simple physical interpretation of <f>. Anj actual state of motion of a liquid, for which a (single-valued) velocity-potential exists, could be produced instantaneously from rest by the appUcation of a properly chosen system of impulsive pressures. This is evident from the equations cited, which shew, moreover, that <^ = mjp + const. ; so that tn = fxf> -\- C gives the requisite system. In the same way w = — p<f> -\- gives the system of impulsive pressures which would completely stop the motion. The occurrence of an arbitrary constant in these expressions shews, what is otherwise evident, that a pressure uniform throughout a liquid mass produces no effect on this motion*. In the case of a gas, <f> may be interpreted as the potential of the extraneous impulsive forces by which the actual motion at any instant could be produced instantaneously from rest. * This interpretation was given by Cauohy, loc, cit., and by Poisson, Mim. deVAcad. roy. des Sciences, t. i. (1816). L.H. 2 18 Integration of the Eqvxztions in Special Cases [chap, ii A state of motion for which a velocity-potential does not exist cannot be generated or destroyed by the action of impulsive pressures, or of extraneous impulsive forces having a potential. 19. The existence of a velocity-potential indicates, besides, certain Hnemalical properties of the motion. A *line of motion' or * stream-line'* is defined to be a hne drawn from point to point, so that its direction is everywhere that of the motion of the fluid. The differential equations of the system of such lines are The relations (1) shew that when a velocity-potential exists the lines of motion are everywhere perpendicular to a system of surfaces, viz. the ' equipotentiar surfaces ff> = const. Again, if from the point (x, y, z) we draw a linear element hs in the direction (Z, w, w), the velocity resolved in this direction \&lu-\- mv •\- nw, or d<f> dx d<f>dy dif>dz i,- i, _ ^^ The velocity in any direction is therefore equal to the rate of decrease of <f> in that direction. Taking hs in the direction of the normal to the surface (f> = const., we see that if a series of such surfaces be drawn corresponding to equidistant values of <f}y the common difference being infinitely small, the velocity at any point will be inversely proportional to the distance between two consecutive surfaces in the neighbourhood of the point. Hence, if any equipotential surface intersect itself, the velocity is zero at the intersection. The intersection of two distinct equipotential surfaces would imply an infinite velocity. 20. Under the circumstances stated in Art. 17, the equations of motion are at once integrable throughout that portion of the fluid mass for which a velocity-potential exists. For in virtue of the relations dv _dw dw ^ du du __ dv dz^ dy* dx~ dz' dy^dx* which are implied in (1), the equations of Art. 6 may be written dxdi dx dx dx "^ dx pdx* '' * Some writers prefer to restrict the use of the term * stream-line' to the case of steady motion, as defined in Art 21. 18-21] Vdodty-Potmtial 19 These have the integral /' ■f-i-a-J«' + m (3) where q denotes the resultant velocity (w* + t;* + y^) j and F(t) is an arbitrary function of t. Our equations take a specially simple form in the case of an incompressible fluid ; viz. we then have l^^-n-\f + F(t), (4) with the equation of continuity a^'^a^'^a?^^' ^^) which is the equivalent of Art. 8 (1). When, as in many cases which we shall have to consider, the boundary conditions are purely kinematical, the process of solution consists in finding a function which shall satisfy (5) and v the prescribed surface-conditions. The pressure p is then given by (4), and is thus far indeterminate to the extent of an additive function of t. It becomes determinate when the value of p at some point of the fluid is given for all values of t. Since the term F (t) is without influence on resultant pressures it is frequently omitted. Suppose, for example, that we have a solid or solids moving through a liquid com- pletely enclosed by fixed boundaries, and that it is possible (t.g, by means of a piston) to apply an arbitrary pressure at some point of the boundary. Whatever variations are made in the magnitude of the force applied to the piston, the motion of the fluid and of the solids will be absolutely unaffected, the pressure at all points instantaneously rising or falling by equal amounts. Physically, the origin of the paradox (such as it is) is that the fluid is treated as absolutely incompressible. In actual liquids changes of pressure are propagated with very great, but not infinite, velocity. K the co-ordinate axes are in motion, the formula for the pressure is -'(^l-|)-«('l-l)-'('|-'g). ■■•■'«' where g* = (w - ii)^ + (v - v)* ^ (w - w)^ (7) This easily follows from the formulae for the accelerations given in Art. 12 (3). Steady Motion. 21. When at every point the velocity is constant in magnitude and direction, i.e. when dt ^' dt "' dt ^' ^^^ everywhere, the motion is said to be * steady.' 2—2 20 Integration of the Equations in Special Cases [chap, ii In steady motion the lines of motion coincide with the paths of the particles. For if P, Q be two consecutive points on a line of motion, a particle which is at any instant at P is moving in the direction of the tangent at P, and will, therefore, after an infinitely short time arrive at Q, The motion being steady, the lines of motion remain the same. Hence the direction of motion at Q is along the tangent to the same line of motion, i.e, the particle continues to describe the line. In steady motion the equation (3) of the last Article becomes dp = - II - iflr2 -i- constant (2) P The law of variation of pressure along a stream-line can however in this case be found without assuming the existence of a velocity-potential. For if hs denote an element of a stream-line, the acceleration in the direction of motion is qdq/dSy and we have dq ^ 9ft 1 dp ^ 3« "" ds pds' whence, integrating along the stream-line, 'dp I- = -ft-igr« + C (3) This is similar in form to (2), but is more general in that it does not assume the existence of a velocity-potential. It must however be carefully noticed that the 'constant' of equation (2) and the 'C of equation (3) have different meanings, the former being an absolute constant, while the latter is constant along any particular stream-line, but may vary as we pass from one stream- line to another. 22. The theorem (3) stands in close relation to the principle of energy. If this be assumed independently, the formula may be deduced as follows*. Taking first the particular case of a hquid, let us consider the portion of an infinitely narrow tube, whose boundary follows the stream-hnes, included between the cross-sections A and J5, the direction of motion being from A to J5. Let p be the pressure, q the velocity, ft the potential of the extraneous forces, a the area of the cross-section, at A, and let the values of the same quantities at B be distinguished by accents. In each unit of time a mass pqa at A enters the portion of the tube considered, whilst an equal mass pq'a leaves it at J5. Hence qa = qW, Again, the work done on the mass entering at ^ is pqa per unit time, whilst the loss of work at J3 is p'((o , The former mass brings with it the energy pqa {\(^ + ft), whilst the latter carries off energy to the amount pjV (Jg'* + ft'). The motion being steady, * This is really a reversion to the methods of Daniel Bernoulli, Hydrodynamica, Argentorati, 1738. 21-23] Steady Motion 21 the portion of the tube considered neither gains nor loses energy on the whole, so that Dividing by pqa (= pq'a), we have 2 + i?^ + n = ^ + i?'* + n', p p or, using C in the same sense as before, J=-ll-iy« + C, (4) which is what the equation (3) becomes when p is constant. To prove the corresponding formula for compressible fluids, we remark that the fluid entering at A now brings with it, in addition to its energies of motion and position, the intrinsic energy per tmit mass. The addition of these terms to (4) gives the equation (3). The motion of a gas is as a rule subject to the adiabatic law P/Po = (p//>o)^ (5) and the equation (3) then takes the form ^jJ = -ft-k« + C (6) 23. The preceding equations shew that, in steady motion, and for points along any one stream-line *, the pressure is, cceteris paribus, greatest where the velocity is least, and vice versa. This statement, though opposed to popular notions, becomes evident when we reflect that a particle passing from a place of higher to one of lower pressure must have its motion accelerated, and vice V€rsd'\. It follows that in any case to which the equations of the last Article apply there is a limit which the velocity cannot exceed f. For instance, let us suppose that we have a liquid flowing from a reservoir where the velocity may be neglected, and the pressure is ^q, and that we may neglect extraneous forces. We have then, in (4), C = pp/p, and therefore P = J>o-iP?' (7) Now although it is found that a Uquid from which all traces of air or other dissolved gas have been eliminated can sustain a negative pressure, or tension, * This restriction is mmeceBsary when a velooity-potential exists. t Some interesting practical illustrations of this principle are given by Fronde, Nature, t. xiii 1875. % Cf. Helmholtz, **Ueber discontinuirliche flassigkeitsbewegnngen/' BerL MonaJUber. April 1868; Pha, Mag, Not. 1808 [WisaenachafiUcke Abhandlungen, Leipzig, 1882-3, t. i. p. 146]. 22 Integration of the Equations in Special Coms [chap, n of considerable magnitude*, this is not the case with fluids such as we find them under ordinary conditions. Practically, then, the equation (7) shews that g cannot exceed (2po/p) • This limiting velocity is of course that with whicb the fluid would escape from the reservoir into a vacuum. In the case of water at atmospheric pressure it is the velocity 'due to' the height of the water-barometer, or about 45 feet per second. If in any case of fluid motion of which we have succeeded in obtaining the analytical expression, we suppose the motion to be gradually accelerated until the velocity at some point reaches the limit here indicated, a cavity will be formed there, and the conditions of the problem are more or less changed. It will be shewn, in the next chapter (Art. 44), that in irrotational motion of a liquid, whether 'steady' or not, the place of least pressure is always at some point of the boundary, provided the extraneous forces have a potential fl satisfying the equation This includes, of course, the case of gravity. In the general case of a fluid in which ;> is a given function of p we have, putting fl = in (3), =2r^ (8) J» p V 9 For a gas subject to the adiabatic law, this gives if c, = iyp/pr, = {dp/dpy, denote the velocity of sound in the gas when at pressure p and density />, and Cq the corresponding velocity for gas under the conditions which obtain in the reservoir. (See Chapter x.) Hence the limiting velocity is 2_\i or2-214co, ify=l-408. 24. We conclude this chapter with a few simple appUcations of the equations. Effiux of Liquids. Let us take in the first instance the problem of the efflux of a Uquid from a small orifice in the walls of a vessel which is kept filled up to a constant level, so that the motion may be regarded as steady. ♦ O. Reynolds, Manch. Mem. t. vL (1877) [Scieniijic Papers, Cambridge, 1900. . ., t. L p. 231]. (, •^o> 23-24] Efflux of Liquids 23 The origin being taken in the upper surface, let the axis of z be vertical, and its positive direction downwards, so that il= — gz. If we suppose the area of the upper surface large compared with that of the orifice, the velocity at the former may be neglected. Hence, determining the value of C in Art. 22 (4) so that p^ P (the atmospheric pressure), when 2 = 0, we have* l^l + gz-^q» (1) r r At the surface of the issuing jet we have p ^^ P, and therefore ?*=2<^, (2) i,e, the velocity is that due to the depth below the upper surface. This is known as TorriceMs Theorem^, We cannot however at once apply this result to calculate the rate of efflux of the fluid, for two reasons. In the first place, the issuing fluid must be regarded as made up of a great number of elementary streams converging from all sides towards the orifice. Its motion is not, therefore, throughout the area of the orifice, everywhere perpendicular to this area, but becomes more and more oblique as we pass from the centre to the sides. Again, the converging motion of the elementary streams must make the pressure at the orifice somewhat greater in the interior of the jet than at the surface, where it is equal to the atmospheric pressure. The velocity, therefore, in the interior of the jet will be somewhat less than that given by (2)*. Experiment shews however that the converging- motion above spoken of ceases at a short distance beyond the orifice, and that (in the case of a circular orifice) the jet then becomes approximately cylindrical. The ratio of the area of the section S' of the jet at this point (called the 'vena contracta') to tbe area S of the orifice is called the 'coefficient of contraction.' If the orifice be simply a hole in a thin wall, this coefficient is found experimentally to be about -62. The paths of the particles at the vena contracta being nearly straight, there is little or no variation of pressure as we pass from the axis to the outer surface of the jet. We may therefore assume the velocity there to be imiform throughout the section, and to have the value given by (2), where z now denotes the depth of the vena contracta below the surface of the Uquid in the vessel. The rate of efflux is therefore {^z)^-pS' (3) The calculation of the form of the issuing jet presents difficulties which have only been overcome in a few ideal cases of motion in two dimensions. (See Chapter iv.) It may however be shewn that the coefficient of con- traction must, in general, he between J and 1. To put the argument in its * ThiB result is due to D. Bernoulli, 2.c. ante p. 20. t '*I>e motu grayium naturaliter accelerato," Firenze, 1643. 24 Integration of the Equations in Special Cases [chap, ii simplest form, let us first take the case of liquid issuing from a vessel the pressure in which, at a distance from the orifice, exceeds that of the external space by the amount P, gravity being neglected. When the orifice is closed by a plate, the resultant pressure of the fluid on the containing vessel is of course nil. If when the plate is removed we assume (for the moment) that the pressure on the walls remains sensibly equal to P, there will be an un- balanced pressure PS acting on the vessel in the direction opposite to that of the jet, and tending to make it recoil. The equal and contrary reaction on the fluid produces in unit time the velocity q in the mass pqS' flowing through the *vena contracta,' whence PS = pqK^' (4) The principle of energy gives, as in Art. 22, P = W, (5) SO that, comparing, we have S' = ^S, The formula (1) shews that the pressure on the walls, especially in the neighbourhood of the orifice, will in reality fall somewhat below the static pressure P, so that the left-hand side of (4) is an under-estimate. The ratio S'/S will therefore in general be > J. In one particular case, viz. where a short cylindrical tube, projecting inwards, is attached to the orifice, the assumption above made is sufficiently exact, and the consequent value | for the coefficient then agrees with experiment. « The reasoning is easily modified so as to take account of gravity (or other conservative forces). We have only to substitute for P the excess of the static pressure at the level of the orifice over the pressure outside. The difference of level between the orifice and the 'vena contracta' is here neglected*. Efflux of Gases. 25. We consider next the efflux of a gas, supposed to flow through a small orifice from a vessel in which the pressure is p^ and density p^ into a space where the pressure is pi. We assume that the motion has become steady, and that the expansion takes place according to the adiabatic law. If the ratio pjpi of the pressures inside and outside the vessel do not exceed a certain limit, to he indicated presently, the flow will take place in much the same manner as in the case of a liquid, and the rate of discharge may be found by putting p =pi in Art. 23 (9), ♦ The above theory is due to Borda {Mim. de VAcad. des Sciences, 1766), who also made experiments with the special form of mouth-piece referred to, and found 818' = 1-942. It was re-discovered by Hanlon, Proa, Lond, Maih, 8oc, t. iii p. 4 (1869); the question is further elucidated in a note appended to this paper by MaxwelL See also Froude and J. Thomson, Proc, Olasgoto Phil. Soc. t. x. (1876). It has been remarked by several writers that in the case of a diverging conical mouth-piece projecting inwards the section at the vena contracta may be less than half the area of the internal orifice. 24-26] Efflux of Oases 25 and multipl3diig the resulting value of q by the area 8' of the vena oontraota. This gives for the rate of discharge of mass * 2 y+1 ,.««'-C4,)'..{(g)'-© ')'•»■■ <■' It is plain, however, that there must be a limit to the applicability of this result ; for otherwise we should be led to the paradoxical conclusion that when py^ =0, i.e. the discharge is into a vacuum, the flux of matter is nil. The elucidation of this point is due to Prof. Osborne Reynolds f- It is easily found by means of Art. 23 (8), that ^p is a maximum, i.e. the section of an elementary stream is a minimum, when g* = dp/dp, that is, the velocity of the stream is equal to the velocity of sound in gas of the pressure and density which prevaO there. On the adiabatic hypothesis this gives, bv Art. 23 ( 10), HM'-- «' and therefore, since c* qc p^ , H^f- ^(4l)'"^ '" or, if y = 1-408, p = -634po. ^ = -527po (4) If Pi be less than this value, the stream -after passing the point in question widens out again, until it is lost at a distance in the eddies due to viscosity. The minimum sections of the elementary streams will be situate in the neighbourhood of the orifice, and their sum J3 may be called the virtual area of the latter. The velocity of efflux, as found from (2), is gr = .911Co. The rate of discharge is then =qp8f where q and p have the values just found, and is there- fore approximately independent of the external pressure Pi so long as this falls below -527 j>0. The ph3r8ical reason of this is (as pointed out by Reynolds) that, so long as the velocity at any point exceeds the velocity of sound under the conditions which obtain there, no change of pressure can be propagated backwards beyond this point so as to affect the motion higher up the stream. These conclusions appear to be in good agreement with experimental results. Under similar circumstances as to pressure, the velocities of efflux of different gases are (so far as y can be assumed to have the same value for each) proportional to the corre- sponding velocities of sound. Hence (as we shall see in Chapter x. ) the velocity of efflux will vary inversely, and the rate of discharge of mass will vary directly, as the square root of the density]:. Rotating Liquid, 26. Let us next take the case of a mass of liquid rotating, under the action of gravity only, with constant and uniform angular velocity co about the axis of z, supposed drawn vertically upwards. * A result equivalent to this was given by Saint VenaDt and Wantzel, Jounu de VicoU PclffL t. xvi p. 92 (1839). t "On the Flow of Gases," Proc. Manch, Lit, and PhU Soc, Nov. 17, 1886; PhiL Mag, March 1886 [Papers, t. it p. 311]. A similar explanation was given by Hugoniot, Comptea Rendus, June 28, July 26, and Dec. 13, 1886. I have attempted, above, to condense the reasoning of these writers. t Cf. Graham, Phil Trans. 1846. 26 Integration of the Eqtuitions in Special Cases [chap, n By hypothesis, w, v, «^ « — aiy, mXy ' 0, Z, Y, Z= 0, 0, -(7. The equation of contiimity is satisfied identically, and the dynamical equations obviously are _„^.-!|, -„v-5|, o.-i|-,. .,» These have the common integral ^ = i<u« (»* + y«) -5^2-1- const (2) P The free surface, p = const., is therefore a paraboloid of revolution about the axis of z, having its concavity upwards, and its latus rectum = 2^/co^. Q. dv du rt Smce 5 ?r- = 2co, ox oy a velocity-potential does not exist. A motion of this kind could not therefore be generated in a 'perfect' fluid, i.e, in one unable to sustain tangential stress. 27. Instead of supposing the angular velocity a> to be uniform, let us suppose it to be a function of the distance r from the axis, and let us inquire what form must be assigned to this function in order that a velocity-potential may exist for the motion. We find dx dy dr* and in order that this may vanish we must have uyr^ «= ft, ft constant. The velocity at any point is then = iilr, so that the equation (2) of Art. 21 becomes ^ = const.-J^, (1) if no extraneous forces act. To find the value of ^ we have, using polar co-ordinates, ar " "' rde" r ' whence ^ = — /xd + const. = — /itan~*- -\- const (2) We have here an instance of a 'cyclic' function. A function is said to be 'single- valued' throughout any region of space when we can assign to every point of that region a definite value of the function in such a way that these values shall form a continuous system. This is not possible with the function (2) ; for the value of ^, if it vary continuously, changes by — 27r/Lt as the point to which it refers describes a complete circuit round the origin. The general theory of cyclic velocity-potentials will be given in the next chapter. 26-28] Rotating Liquids 27 If gravity act, and if the axis of z be drawn vertically upwards, we must add to (1) the term — gz. The form of the free surface is therefore that generated by the revolution of the hyperbolic curve xH = const, about the axis of z. By properly fitting together the two preceding solutions we obtain the case of Rankine's 'combined vortex.' Thus the motion being everywhere in coaxial circles, let us suppose the velocity to be equal to wr from r = to r == a, and to cja^r for r> a. The corresponding forms of the free surface are then given by ' and 2 = ^(r*-O«) + 0, these being continuous when r = a. The depth of the central depression below the general level of the surface is therefore wl^a'^jg. 28. To illustrate, by way of contrast, the case of extraneous forces not having a potential, let us suppose that a mass of liquid filling a right circular cylinder moves from rest under the action of the forces X^Ax + By, Y = B'x^Cy, Z«0, the axis of z being that of the cylinder. If we assume u = - ^y, t; = taXy u? =0, where o> is a function of t only, these values satisfy the equation of continuity and the boundary conditions. The dynamical equations are evideiitly dt pox d» \dp (1) Differentiating the first of these with respect to y, and the second with respect to x and subtracting, we eliminate p, and find J=i(B'-*) (2) 28 Integration of the Eqtuitions in Special Cases [chap, n The fluid therefore rotates as a whole about the axis of z with constantly accelerated angular velocity, except in the particular case when B=B\ To find p, we substitute the value of da>/dt in (1) and integrate; we thus get ? =i<»* (x* +y*) +i (-4a?« +2/3xy +0y«) +const., P where 2p=B+B\ 29. Afl a final example, we will take one suggested by the theory of 'electro-magnetic rotations.' If an electric current be made to pass radially from an axial wire, through a conducting liquid, to the walls of a metallic containing cylinder, in a uniform magnetic field, the extraneous forces will be of the type* Assuming tt = - ay, v —mx, u; =0, where o) is a function of r and t only, we have Bq> , uX \dp P^' =0. («) Eliminating p, we obtain The solution of this is <a=F (t)/r* +/ (r), where F and / denote arbitrary functions. Since » =0 when ^ =0, we have i^(0)/r*+/(r)=0, andtherefore ^^ J^ffl - J^(0) ^ X where X is a function of t which vanishes for ^ =0. Substituting in (1), and integrating, we find Since p is essentially a single- valued function, we must have dK/dt =/ui, or X =/i/. Hence the fluid rotates with an angular velocity which varies inversely as the square of the distance from the axis, and increases constantly with the time. * It C denote the total flux of electricity outwards, per unit length of the axis, and y the component of the magnetic force parallel to the axis, we have fi=yCl2wp. For the history of such experiments see Winkelmann, Handlmch d. Physik, 2nd ed. (1905), t. v. (1), p. 430. The above case is specially simple, in that the forces X, F, Z have a potential (0= -m tan~^ yl^)» though a * cyclic' one. As a rule, in electro-magnetic rotations, the mechanical forces X, Y, Z have not a potential at all. CHAPTER III IRROTATIONAL MOTION 30. The present chapter is devoted mainly to an exposition of some general theorems relating to the kinds of motion already considered in Arts. 17 — 20; viz. those in which udx-^vdy + wdz is an exact differential throughout a finite mass of fluid. It is convenient to begin with the following analysis, due to Stokes*, of the motion of a fluid element in the most general case. The component velocities at the point (x, y, z) being i«, v, tr, the relative velocities at an infinitely near point (x + 8x, y + 8y, z + hz) are ^ du ^ , du ^ , du ^ ^''^di^'^-^'dy^y^dz^'^ dx ^ dy ^ ^ dz ' 5 dw^ , dw ^ , dw^ ^"di^ + d^^'J+Tz^'-^ (1) If we write du , dv dw '^dz' - dw dv ^~ dy^dz' du dw ^^ dz^ dx* , dv du dx dy* ^ dw dv ^"dy dz' du dw "^^dz dx' y dv du. equations (1) may be written 8w = a8x + \l&y + \glz -h \ {r^z - ^y)\ 8t;= iA8a;+ % + i/Sz + i(C8x - f8z),l (2) 8w = \ghx + \Jhj + cSz + \{&y - lySx). J * *'0n the Theories of the Internal Friction of Fluids in Motion, &c." CwnA, Phil. Trans. t. Tiii. (1845) [Papers, t. i. p. 80]. t There is here a deviation from the traditional convention. It has been customary to use symbols such as {, 17, ^ (Helmholtz) or io\ ia\ ta'" (Stokes) to denote the component rotations 1 ^dw dv' 2 of a fluid element. The fundamental kinematical theorem is however that of Art. 32 (3), and the definition of (, 17, ^ now adopted in the text avoids the intrusion of an unnecessary factor 2 (or i as the case may be) in this and in a whole series of subsequent formulae relating to vortex motion. It also improves the electro-magnetic analogy of Art. 148. Xdy'dzJ' iKdz'dz)' 2\dx dy) 30 Irrotational Motion [chap, m Hence the motion of a small element having the point (x, y, z) for its centre may be conceived as made up of three parts. The first part, whose components are u, t?, tc;, is a motion of transUuion of the element as a whole. The second part, expressed by the first three terms on the right-hand sides of the equations (2), is a motion such that, if Sx, 8^, 82; be regarded as current co-ordinates, every point is moving in the direction of the normal to that quadric of the system a {hxY + b {hyY + c {hzf +f8y8z -\- gSzSx + h8x8y = const., ... (3) on which it lies. If we refer these quadrics to their principal axes, the corresponding parts of the velocities parallel to these axes will be 8u' = a'Bx\ 8v' = V8y\ 8w' = c'8z\ (4) if a' (8x')2 + 6' (8y')* + C (Sz')* = const. is what (3) becomes by the transformation. The formulae (4) express that the length of every line in the element parallel to x' is being elongated at the (positive or negative) rate a\ whilst Unes parallel to y' and z' are being elongated in like manner at the rates b' and c' respectively. Such a motion is called one of vure strain a nd the principal axes of the quadrics (3) are called the axes of the strain. The last two terms on the right-hand sides of the equations (2) express a rotation of the element as a whole about an instantaneous axis; the com- ponent angular velocities of the rotation being J^, Jt;, J{*. The vector whose components are ^, ly, £ may conveniently be called the ' vorticity' of the medium at the point {x, y, z). This anaiysis may be iUustrated by the so-called * laminar* motion of a liquid. Thus if we have a, b, c, /, g, (, rj =0, h =y^ f = -/*. If A represent a rectangular fluid element bounded by planes parallel to the co-ordinate planes, then B represents the change produced in this in a short time by the strain alone, and C that due to the strain pl%i8 the rotation. rr y' * The quantities oorresponding to \^, {yi, J^in the theory of the infinitely small displacemenia of a continuous medium had been interpreted by Cauchy as expressing the *mean rotations* of an element, Exercices d* Analyse et de Physique, t. ii. (Paris, 1841), p. 302. 30-81] Deformation of an Element 31 It is easily seen that the above resolution of the motion is unique. If we assume that the motion relative to the point (a;, y, z) can be made up of a strain and a rotation in which the axes and coefficients of the strain and the axis and angular velocity of the rotation are arbitrary, then calculating the relative velocities 8u, 8r, hwy we get expressions similar to those on the right- hand sides of (2), but with arbitrary values of a, 6, c,/, g,h,(, 17, J. Equating coefficients of 8a;, by, 8z, however, we find that a, b, c, Sec. must have respec- tively the same values as before. Hence the directions of the axes of the strain, the rates of extension or contraction along them, and the axis and the amount of the vorticity, at any point of the fluid, depend only on the state of relative motion at that point, and not on the position of the axes of reference. When throughout a finite portion of a fluid mass we have ^, '7, C &11 ^^^o, the relative motion of any element of that portion consists of a pure strain only, and is called 'irrotational.' 31. The value of the integral J {udx + vdy + wdz). or ff dx , dy ^ dz\ , taken along any line ABCD, is called* the 'flow' of the fluid from A to D along that line. We shall denote it for shortness by / (ABCD). If A and D coincide, so that the line forms a closed curve, or circuit, the value of the integral is called the 'circulation' in that circuit. We denote it by / (ABCA). If in either case the integration be taken in the opposite direction, the signs of dx/ds, dy/ds, dz/ds will be reversed, so that we have I (AD) ^ - I (DA), and I (ABCA) = ^ I (ACBA). It is also plain that / (ABCD) = / (AB) + / (BG) -h / (CD). Again, any surface may be divided, by a double series of lines crossing it, into infinitely small elements. The sum of the circulations round the boundaries of these elements, taken all in the same sense, is equal to the circulation round the original boundary of the surface (supposed for the moment to consist of a single closed curve). For, in the sum in question, the flow along each side common to two elements comes in twice, once for each element, but with opposite signs, and there- fore disappears from the result. There remain then only the flows along those sides which are parts of the original boundary; whence the truth of the above statement. ♦ Sir W. Thomson, "On Vortex Motion," Edin, Trans, t. xxv. (1869) [Papers, L iv. p, 13]. 32 Irrotational Motion [chap, m From this it follows, by considerations of continuity, that the circulation round the boundary of any surface-element hS, having a given position and aspect, is ultimately proportional to the area of the element. If the element be a rectangle 8^ hz having its centre at the point (x, y, z), then calculating the circulation round it in the direction shewn by the arrows in the annexed figure, we have y and therefore I (AB) = {v - J {dv/dz) Sz} 8y, / (BC) = {w + i {dw/dy) Sy} Sz, I (CD) = - {v + J (dv/dz) 8z} Sy, I (DA) = - {w? - i (dw/dy) hy} Sz, In this way we infer that the circulations round the boundaries of any infinitely small areas 88 1, SS^, 8/S3, having their planes parallel to the co-ordinate planes, are ^8Si, 7y8S„ C853 (1) respectively. Again, referring to the figure and the notation of Art. 2, we have / (ABC A) = / (PBCP) + / (PCAP) + / (PABP) = f . iA + ^ . niA + f . wA, whence we infer that the circulation round the boundary of any infinitely small area 88 is {l^-\-m7i + n088 (2) We have here an independent proof that the quantities ^, ly, C ^^7 be regarded as the components of a vector. It will be observed that some convention is implied as to the relation between the sense in which the circulation round the boundary of 85 is estimated, and the sense of the normal (2, m, n). In order to have a clear understanding on this point, we shall suppose in this book that the axes of co-ordinates form a right-handed system ; thus if the axes of x and y point E. 31-32] Cireulation in a Finite Circuit 33 and K. respectively, that of a will point vertically upwards*. The aeiiBe in which the circulation, as given by (2), is estimated is then related to the ditection of the normal {I, m, n) in the mannei typified by a ligbt-handed screw t- 32. Expressing now that the circulation round the edge of any finite surface Js equal to the sum of the circulations round the boundaries of the infinitely small elements into which the surface mav be divided, we have, by (2), ; (uit + sij + »&)-;; (if + m, + n{) (B (3) or, substituting the values of f , ij, i from Art. 30, '<"*--*+-^)=//KI-l)+-(s-s)+«(£-|)H--'*> ■ where the single-integral is taken along the bounding curve, and the donble- integral over the surface :[■ ^° these formulae the quantities I, m, n are the direction-cosines of the normal drawn always on one side of the surface, which we may term the positive aide; the direction of integration in the first member is then that in which a man walking on the surface, on the positive side of it, and close to the edge, must proceed so as to have the surface always on his left hand. The theorem (3) or (4) may evidently be extended to a surface whose boundary consists of two or more closed curves, provided the integration in the first member be taken round each of these in the proper direction, according to the rule just given. Thus, if the surface-integral in (4) extend over the shaded portion of the annexed figure, the directions in which the circulations in the several parts of the boundary are to be taken are shewn by the arrows, the positive side of the surface being that which faces the reader. The value of the surface-integral taken over a closed surface is zero. It should be noticed that (4) is a theorem of pure mathematics, and is true whatever functions u, v, w may be of x, y, z, provided only they be continuous and difierentiable at all points of the surface§. * Huwell. Proc. Load. MaOi. Soe. t. iiL pp. 279, 280. Thus in the fig. of p. 32 th« axis tit X is Bopposed drawn towuds the reader. t Sse HaxweU, EUetrieily and MagiKtUm, Oxford, 1873, Art 23. X Thia tlieorem b due to Stokes, Smith's Prize Examinatiim Paper* /or 1864. the fiiBt pobliahed proof Appears to have been given by Hanbel, Zur aUftm. Tkeorie der Snee^n; der FlStngkeiUn, Gottingen, 1861, p. 39. That given above ie due to Lord Kelvin, tc onle p. 31. See ako Thomson and Tait, Nafural Philo»opky, Art. 190 (», and Maxwell, EUctrieity and Magntlittn, Art. 24. J It ia not neoesBary that their di&erential <<oefficient« shoald bi 34 Irrotaiional Motion [chap, m 33. The rest of this chapter is devoted to a study of the kinematical properties of irrotational motion in general, as defined by the equations f,^,C = 0, (1) i.e. the circulation in every infinitely small circuit is assumed to be zero. The existence and properties of the velocity-potential in the various cases that may arise will appear as consequences of this definition. The phjrsical importance of the subject rests on the fact that if the motion of any portion of a fluid mass be irrotational at any one instant it will under certain very general conditions continue to be irrotational. Practically, as will be seen, this has already been established by Lagrange's theorem, proved in Art. 17, but the importance of the matter warrants a repetition of the investigation, in terms of the Eulerian notation, in the form given by Lord Kelvin*. Consider first any terminated line AB drawn in the fluid, and suppose every point of this line to move always with the velocity of the fluid at that point. Let us calculate the rate at which the flow along this line, from A to B, is increasing. If 8x, Sy, 8z be the projections on the co-ordinate axes of an element of the line, we have -(uSx)=-^8x + u-j^. Now DSx/Dt, the rate at which Sx is increasing in consequence of the motion of the fluid, is equal to the difference of the velocities parallel to x at the two ends of the element, i,e, to 8u ; and the value of Du/Dt is given by Art. 5. Hence, and by similar considerations, we find, if p be a function of p only, and if the extraneous forces X, Y, Z have a potential H, =r- (u8x + vSv + w8z) = — ^ — 8n -i- u8u -h v8v + t(f8w. Dt "^ p Integrating along the line, from ^4 to 5, we get ^W^ ^|^(w(fa + vdy+M;(iz)=|^-|^-n4-i?* \ (2) or, the rate at which the flow from il to jB is increasing is equal to the excess of the value which — Sdp/p — fl + i?* has at B over that which it has at A, This theorem comprehends the whole of the dynamics of a perfect fluid. For instance, equations (2) of Art. 15 may be derived from it by taking as the line AB the infinitely short line whose projections were originally 8a, 86, 8c, and equating separately to zero the coeflScients of these infinitmmals. If ft be single-valued, the expression within brackets on the right-hand side of (2) is a single-valued function of x, y, z. Hence if the integration on * Ic. ante p. 31. 33-35] Vdoeity-Fotential 35 the left-hand be taken round a closed curve, bo that B coincides with A, we have D f yjT I {udx + vdy + wdz) = 0, (3) Dt or, the circulation in any circuit moving with the fluid does not alter with the time. It follows that if the motion of any portion of a fluid mass be initially irrotational it will always retain this property ; for otherwise the circulation in every infinitely small circuit would not continue to be zero, as it is initially by virtue of Art. 32 (3). 34. Considering now any region occupied by irrotationally-moving fluid, we see from Art. 32 (3) that the circulation is zero in every circuit which can be filled up by a continuous surface lying wholly in the region, or which in other words is capable of being contracted to a point without passing out of the region. Such a circuit is said to be 'reducible.' Again, let us consider two paths ACB, ADB, connecting two points A, B of the region, and such that either may by continuous variation be made to coincide with the other, without ever passing out of the region. Such paths are called 'mutually reconcileable.' Since the circuit ACBDA is reducible, we have / {ACBDA) = 0, or since / (BDA) = - / (ADB), I (ACB) ^ I (ADB); i.e. the flow is the same along any two reconcileable paths. A region such that aU paths joining any two points of it are mutually reconcileable is said to be 'simply-connected.' Such a region is that enclosed within a sphere, or that included between two concentric spheres. In what follows, as far as Art. 46, we contemplate only simply-connected regions. 35. The irrotational motion of a fluid within a simply-connected region is characterized by the existence of a single-valued velocity-potential. Let us denote by — ^ the flow to a variable point P from some fixed point A, viz. ^ = — I {udx -{- vdy -^ wdz) (1) The value of <f> has been shewn to be independent of the path along which the integration is effected, provided it lie wholly within the region. Hence ^ is a single- valued function of the position of P ; let us suppose it expressed in terms of the co-ordinates (x, y, z) of that point. By displacing P through an infinitely short space parallel to each of the axes of co-ordinates in succession, we find »-£■ "--| "-I '^' ■ i,e. ^ is a velocity-potential, according to the definition of Art. 17. 3—2 36 Irrotational Motion [chap, in The substitution of any other point B for A, as the lower limit of the integral in (1), simply adds an arbitrary constant to the value of <f>, viz. the flow from A to B. The original definition of <f> in Art. 17, and the physical interpretation in Art. 18, alike leave the function indeterminate to the extent of an additive constant. As we follow the course of any line of motion the value of (f> continually decreases; hence in a simply-connected region the lines of motion cannot form closed curves. 36. The function </> with which we have here to do is, together with its first differential coefficients, by the nature of the case, finite, continuous, and single-valued at all points of the region considered. In the case of incom- pressible fluids, which we now proceed to consider more particidarly, <f} must also satisfy the equation of continuity, (5) of Art. 20, or as we shall in future write it, for shortness, VV = 0, (1) at every point of the region. Hence ^ is now subject to mathematical conditions identical with those satisfied by the potential of masses attracting or repelling according to the law of the inverse square of the distance, at all points external to such masses; so that many of the results proved in the theories of Attractions, Electrostatics, Magnetism, and the Steady Flow of Heat, have also a hydrodynamical application. We proceed to develop those which are most important from this point of view. In any case of motion of an incompressible fluid the surface-integral of the normal velocity taken over any surface, open or closed, is conveniently called the *flux' across that surface. It is of course equal to the volume of fluid crossing the surface per unit time. When the motion is irrotational, the flux is given by d<f> -I! dn^' where SS is an element of the surface, and Sn an element of the normal to it^ drawn in the proper direction. In any region occupied whoUy by liquid, the total flux across the boundary is zero, i.e. II &■«»-». (2) the element 8w of the normal being drawn always on one side (say inwards), and the integration extending over the whole boundary. This may be regarded as a generaUzed form of the equation of continuity (1). The lines of motion drawn through the various points of an infinitesimal circuit define a tube, which may be called a tube of flow. The product of the velocity (q) into the cross-section (a, say) is the same at all points of such a tube. 35-38] Ttibes of Flow 37 We may, if we choose, regard the whole space occupied by the fluid as made up of tubes of flow, and suppose the size of the tubes so adjusted that the product qa is the same for eiach. The flux across any surface is then proportional to the number of tubes which cross it. If the surface be closed, the equation (2) expresses the fact that as many tubes cross the surface inwards as outwards. Hence a line of motion cannot begin 'or end at a point of the fluid, 37. The function <f} cannot be a maximum or a minimum at a point in the interior of the fluid ; for, if it were, we should have d{f>/dn everywhere positive, or everywhere negative, over a small closed surface surrounding the point in question. Either of these suppositions is inconsistent with (2). Further, the square of the velocity cannot be a mcucimum at a point in the interior of the fluid. For let the axis of x be taken parallel to the direction of the velocity at any point P. The equation (1), and therefore abo the equation (2), is satisfied if we write d<f)/dx for </>, The above argument then shews that d<f>/dx cannot be a maximum or a minimum at P. Hence there must be points in the immediate neighbourhood of P at which (3^/3aj)* and therefore a fortiori (S)'+(l)'+^^' ,dyj \dzj is greater than the square of the velocity at P*. On the other hand, the square of the velocity may be a minimum at some point of the fluid. The simplest case is that of a zero velocity; see, for example, the figure of Art. 69, below. 38. Let us apply (2) to the boundary of a finite spherical portion of the liquid. If r denote the distance of any point from the centre of the sphere, hm the elementary solid angle subtended at the centre by an element 8S of the surface, we have d(f}/dn = — d<f>/dr, and 85 = r^to. Omitting the factor r*, (2) becomes |<i»-0, // or ^jj<f>dm = (3) Since l/iir . ff<f}dm or l/47rr* - Si<f>dS measures the mean value of </> over the surface of the sphere, (3) shews that this mean value is independent of * This theorem was enunciated, in another connection, by Lord Kelvin, Phil. Mag, Oct. 1S60 [Reprint of Papers an Electrogtalies, dtc, London, 1872, Art, 606]. The above demon- stration is due to Kirchho£f, Vorlesungen Hher mathemaiiwhe Phyeik, Meehanik^ Leipzig, 1876, p. 186. For another proof see Art. 44 below. 38 Irrotdtional Motion [chap, in the radius. It is therefore the same for any sphere, concentric with the former one, which can be made to coincide with it by gradual variation of the radius, without ever passing out of the region occupied by the irrotationally moving liqui<i. We may therefore suppose the sphere contracted to a point, and so obtain a simple proof of the theorem, first given by Gauss in his memoir* on the theory of Attractions, that the mean value of ^ over any spherical surface throughout the interior of which (1) is satisfied, is equal to its value at the centre. The theorem, proved in Art. 37, that <f> cannot be a maximum or a minimum at a point in the interior of the fluid, is an obvious consequence of the above. The above proof appears to be due, in principle, to Frost f. Another demonstration, somewhat different in form, has been given by Lord Rayleighij:. The equation (1), being linear, will be satisfied by the arithmetic mean of any number of separate solutions ^x, ^2' ^s* ^^ ^ suppose an infinite number of systems of rectangular axes to be arranged uniformly about any point P as origin, and let <f>i,<f>29<f>3f • • - ^^ the velocity-potentials of motions which are the same with respect to these several systems as the original motion <f> is with respect to the system a;, y, z. In this case the arithmetic mean (^, say) of the functions <f>i, ^2> ^3' * • * ^^^ ^^ ^ function of r, the distance from P, only. Expressing that in the motion (if any) represented by 4>i the flux across any spherical surface which can be contracted to a point, without passing out of the region occupied by the fluid, would be zero, we have ^.'*.o. or ^ = const. 39. Again, let us suppose that the region occupied by the irrotationally moving fluid is *periphractic,'§ i.e. that it is limited internally by one or more closed surfaces, and let us apply (2) to the space included between one (or more) of these internal boundaries, and a spherical surface completely enclosing it (or them) and Ijring wholly in the fluid. If M denote the total flux into this region, across the internal boundary, we find, with the same notation as before. // ^iS.-U. * "Allgemeine Lehrsatze, u.8.w.," ResuUaie aua den Bedbachtungen des magnetischen Vereins, 1839 [Werke, Qottingen, 1870-80, t. v. p. 199]. t Quarterly Journal of MaiJiematieaf t. zii (1873). t Messenger of Maihematics, t. vii p. 69 (1878) [ScierUific Papers, Cambridge, 1899. . ., t. L p. 347]. § See Maxwell, Electricity and Magnetism, Arte. 18, 22. A region is said to be 'aperiphractic* when every closed suiface drawn in it can be contracted to a point without passing out of the region. 38-40] Mean Value over a Spherical Surface 39 the suifaoe-integrsl extending over the sphere only. This may be written M ^I.ff4,dm = - 4ffr»' whence ^J|^eiS = ^//^de = ^+ C (4) That is, the mean value of <f} over any spherical surface drawn under the above-mentioned conditions is equal to M/4^rrr + C, where r is the radius, M an absolute constant, and C a quantity which is independent, of the radius but may vary with the position of the centre*. If however the original region throughout which' the irrotational motion holds be unlimited externally, and if the first derivative (and therefore all the higher derivatives) of <f> vanish at infinity, then C is the same for all spherical surfaces enclosing the whole of the internal boundaries. For if such a sphere be displaced parallel to xf, without alteration of size, the rate at which C varies in consequence of this displacement is, by (4), equal to the mean value of d<f)/dx over the surface. Since d<f>/dx vanishes at infinity, we can by taking the sphere large enough make the latter mean value as small as we please. Hence C is not altered by a displacement of the centre of the sphere parallel to X. In the same way we see that C is not altered by a displacement parallel to y or 2; i.e. it is absolutely constant. If the internal boundaries of the region considered be such that the total flux across them is zero, e.g. if they be the surfaces of solids, or of portions of incompressible fluid whose motion is rotational, we have if = 0, so that the mean value of <f> over any spherical surface enclosing them all is the same. 40. (a) If <f> be constant over the boundary of any simply-connected region occupied by liquid moving irrotationally, it has the same constant value throughout the interior of that region. For if not constant it would necessarily have a maximum or a minimum value at some point of the region. Otherwise : we have seen in Arts. 35, 36 that the lines of motion cannot begin or end at any point of the region, and that they cannot form closed curves Ijdng wholly within it. They must therefore traverse the region, beginning and ending on its boundary. In our case however this is impossible, for such a line always proceeds from places where <f> is greater to places where it is less. Hence there can be no motion, i.e. d<f> o<p d(f> -. dx' 8^' di^ ' and therefore <f> is constant and equal to its value at the boundary. ^ It is imdentoody of course, that the spherical surfaces to which this statement applies are reconoileable with one another, in a sense analogous to that of Art. 34. t Kirchhofif, Meehanik, p. 191. 40 IrrotcUional Motion [chap, in (P) Again, if d^jdn be zero at every point of the boundary of such a region as is above described, <f> will be constant throughout the interior. For the condition d^jdn = expresses that no lines of motion enter or leave the region, but that they are all contained within it. This is however, as we have seen, inconsistent with the other conditions which the lines must conform to. Hence, as before, there can be no motion, and (f> is constant. This theorem may be otherwise stated as follows : no continuous irrota- tional motion of a liquid can take place in a simply-connected region bounded entirely by fixed rigid walls. (y) Again, let the boundary of the region considered consist partly of surfaces 8 over which <f> has a given constant value, and partly of other surfaces 2 over which d<f>/dn = 0. By the previous argument, no lines of motion can pass from one point to another of S, and none can cross S. Hence no such lines exist ; <f> is therefore constant as before, and equal to its value at/S. It follows from these theorems that the irrotational motion of a liquid in a simply-connected region is determined when either the value of <f>, or the value of the inward normal velocity — d<f>/dn, is prescribed at aU points of the boundary, or (again) when the value of <f> is given over part of the boundary, and the value of — d<^/dn over the remainder. For ii<f}j^, <f>2 be the velocity- potentials of two motions each of which satisfies the prescribed boundary- conditions, in any one of these cases, the functional — <f>2 satisfies the condition (a) or (P) or (y) of the present Article, and must therefore be constant throughout the region. 41. A glass of cases of great importance, but not strictly included in the scope of the foregoing theorems, occurs when the region occupied by the irrotationally moving liquid extends to infinity, but is bounded internally by one or more closed surfaces. We assume, for the present, that this region is simply-connected, and that <f> is therefore single- valued. If if) be constant over the internal boundary of the region, and tend every- where to the same constant value at an infinite distance from the internal boundary, it is constant throughout the region. For otherwise <f> would be a maximum or a minimum at some point within the region. We. infer, exactly as in Art. 40, that if <f> be given arbitrarily over the internal boundary, and have a given constant value at infinity, its value is everywhere determinate. Of more importance in our present subject is the theorem that, if the normal velocity be zero at every point of the internal boundary, and if the fluid be at rest at infinity, then <f} is everywhere constant. We cannot how- ever infer this at once from the proof of the corresponding theorem in Art. 40. It is true that we may suppose the region limited externally by an infinitely 40-41] Conditions of Determinateness 41 large surface at every point of which d<f}/dn is infinitely small; but it is conceivable that the integral i[d<f>jdn . dS, taken over a portion of this surface, might still be finite, in which case the investigation referred to would fail. We proceed therefore as follows. Since the velocity tends to the limit zero at an infinite distance from the internal boundary (£», say), it must be possible to draw a closed surface S completely enclosing S, beyond which the velocity is everywhere less than a certain value e, which value may, by making IE large enough, be made as 9mall as we please. Nov in any direction from S let us take a point P at such a distance beyond % that the soUd angle which S subtends at it is infinitely small ; and with P as centre let us describe two spheres, one just excluding, the other just including S, We shall prove that the mean value of <f> over each of these spheres is, within an infinitely small amount, the same. For if Q, Q' be points of these spheres on a common radius PQQ\ then if Q, Q' fall within 2 the corresponding values of </> may differ by a finite amount ; but since the portion of either spherical surface which falls within S is an infinitely small fraction of the whole, no finite difference in the mean values can arise from this cause. On the other hand, when Q, Q' fall without 2, the corresponding values of <f> cannot differ by so much as € . QQ\ for € is by definition a superior limit to the rate of variation of ^. Hence, the mean values of <f> over the two spherical surfaces must differ by less than € . QQ\ Since QQ' is finite, whilst € may by taking 2 large enough be made as small as we please, the difference of the mean values may, by taking P sufficiently distant, be made infinitely small. Now we have seen in Arts. 38, 39 that the mean value of <f> over the inner sphere is equal to its value at P, and that the mean value over the outer sphere is (since ilf = 0) equal to a constant quantity C. Hence, ultimately, the value of i at infinity tends everywhere to the constant value C. The same result holds even if the normal velocity be not zero over the internal boundary ; for in the theorem of Art. 39 ilf is divided by r, which is in our case infinite. It follows that if d<f>ldn = at all points of the internal boundary, and if %he fluid be at rest at infiinity, it must be everywhere at rest. For no lines of motion can begin or end on the internal boundary. Hence such lines, if they existed, must come from an infinite distance, traverse the region occupied by the fluid, and pass off ag^in to infinity; i,e, they must form infinitely long courses between places where <f> has, within an infinitely small amount, the same value C, which is impossible. The theorem that, if the fluid be at rest at infinity, the motion is deter- minate when the value of — 3^/9n is given over the internal boundary, follows by the same argument as in Art. 40. 42 Irrotational Motion [chap, m Green's Theorem. 42. In treatises on Electrostatics, &c., many important properties of the potential are usually proved by means of a certain theorem due to Green. Of these the most important from our present point' of view have already been given ; but as the theorem in question leads, amongst other things, to a useful expression for the kinetic energy in any case of irrotational motion, some account of it will properly find a place here. Let U, F, W be any three functions which are finite, single-valued and differentiable at all points of a connected region completely bounded by one or more closed surfaces S ; let 8S be an element of any one of these surfaces, and I, m, n the direction-cosines of the normals to it drawn inwards. We shall prove in the first place that ffilU + mV + nW)dS^ "///(^ "*" 1^ + ^)dxdydz, ... .(1) where the triple-integral is taken throughout the region, and the double- integral over its boundary. If we conceive a series of surfaces drawn so as to divide the re^on into any number of separate parts, the integral saw -{-mV + nW)dS, ,..,(2) taken over the original boundary, is equal to the sum of the similar integrals each taken over the whole boundary of one of these parts. For, for every element Sa of a dividing surface, we have, in the integrals corresponding to the parts lying on the two sides of this surface, elements {lU + mV + vbW) 8a, and {VU + m^V + n'W) 8<7, respectively. But the normals to which ?, m, n and V, m\ n' refer being drawn inwards in each case, we have Z' = — Z, w' = — w, 7(' = — n ; so that, in forming the sum of the integrals spoken of, the elements due to the dividing surfaces disappear, and we have left only those due to the original boundary of the region. Now let us suppose the dividing surfaces to consist of three systems of planes, drawn at infinitesimal intervals, parallel to yZy zx, xy, respectively. If X, y, z be the co-ordinates of the centre of one of the rectangular spaces thus formed, and 8x, Sy, Sz the lengths of its edges, the part of the integral (2) due to the yz-iskoe nearest the origin is and that due to the opposite face is The sum of these is —dU/dx ,8xSy8z. Calculating in the same way the 42-43] Greeii's Theorem 43 parts of the integral due to the remaining pairs of faces, we get for the final result Hence (1) simply expresses the fact that the surface-integral (2), taken over the boundary of the region, is equal to the sum of the similar integrals taken over the boundaries of the elementary spaces of which we have supposed it built up. It is evident from (1), or it may be proved directly by transformation of co-ordinates, that if ?7, F, TT be regarded as components of a vector, the expression ^jo_ ^y dw_ dx dy dz is a 'scalar' quantity, i.e. its value is unaffected by any such transformation. It is now usually called the * divergence' of the vector-field at the point (Xy y, z). The interpretation of (1), when (Z7, F, W) is the velocity of a continuous substance, is obvious. In the particular case of irrotational motion we obtain jj^dS=-IJfvhl,dxdydz (3) where Sn denotes an element of the inwardly-directed normal to the surface S. Again, if we put U, Vj W ^ pUy pv, pw, respectively, we reproduce in substance the second investigation of Art. 7. Another useful result is obtained by putting ?7, F, TF = u<f>, t^, w<f}, respectively, where w, v, w satisfy the relation 3i* 9t; 3tt? _ ^ dx dy dz throughout the region, and make lu + mv + nw = over the boundary. We find ///(»! +"i +"!)*'**-»• w The function <f> is here merely restricted to be finite, single- valued, and con- tinuous, and to have its first differential coefficients finite, throughout the region. 43. Now let <f>, <f>^ be any two functions which, together with their first and second derivatives, are finite and single-valued throughout the region considered; and let us put respectively, so that lU + mV + nW = ^ -^^ 44 Irrotational Motion [chap, m $ub6titutmg in (1) we find - lll^^^' dxdydz (5) By interchanging ^ and <f> we obtain -- lll<f>'Vhf>dxdydz (6) Equations (5) and (6) together constitute Green's theorem*. 44. If <f>y if}' be the velocity-potentials of two distinct modes of irrotational motion of a Uquid, so that VV = 0, V^' = 0, (1) we ob tain jh^^'^^j^'^^ ^^ ^ If we recall the physical interpretation of the velocity-potential, given in Art. 18, then, regarding the motion as generated in each case impulsively from rest, we recognize this equation as a particular case of the dynamical theorem that where p^, qr and p/ , q/ are generalized components of impulse and velocity, in any two possible motions of a system f. Again, in Art. 43 (6) let </>' = ^, and let <f} be the velocity-potential of a liquid. We obtain To interpret this we multiply both sides by J/o. Then on the right-hund side — d<f}/dn denotes the normal velocity of the fluid inwards, whilst p(f} is, by Art. 18, the impulsive pressure necessary to generate the motion. It is a proposition in Dynamics j: that the work done by an impulse is measured by the product of the impulse into half the sum of the initial and final velocities, resolved ip. the direction of the impulse, of the point to which it is applied. Hence the right-hand side of (3), when modified as described, expresses the work done by the system of impulsive pressures which, applied to the surface S, would generate the actual motion; whilst the left-hand side gives the kinetic energy of this motion. The formula asserts that these two quantities * 0. Green* EsMy on Electricity and Magnetism, Nottingham, 1828, Art. 3 [McUhematical Papers (ed. Ferreta), Cambridge, 1871, p. 23]. t Thomson and Tait, Natural PhUoeophy, Art. 313, equation (11). t Ibid, Art. 308. 43-45] Kinetic Energy 45 are equal. Hence if T denote the total kinetic energy of the liquid, we have the very important result 2^=-^/Mt'^ ••••• w If in (3), in place of <^, we write dfftjdx, which will of course satisfy V*9^/3x=0, and apply the resulting theorem to the region included within a spherical surface of radius r having any point (x, y, z) as centre, then with the same notation as in Art. 39, we have Hence, writing q* = u^ + v^+w^. Since this latter expression is essentially positive, the mecin value of g', taken over a sphere having any given point as centre, increases with the radius of the sphere. Hence 9* cannot he a maximum at any poiAt of the fluid, as was proved otherwise in Art. 37. Moreover, recalling the formula for the pressure in any case of irrotational motion of a liquid, viz. ?=|_O-l^.i-(0 (6) we infer that, provided the potential Q of the external forces satisfy the condition V«Q=0 (7) the mean value of p over a sphere described with any point in the interior of the fluid as centre will diminish as the radius increases. The place of least pressure will therefore be somewhere on the boundary of the fluid. This has a bearing on the point discussed in Art. 23. 45. In this connection we may note a remarkable theorem discovered by Lord Kelvin*, and afterwards generalized by him into an universal property of dynamical systems started impulsively from rest under prescribed velocity- conditions f. The irrotational motion of a liquid occupying a simply-connected region has less kinetic energy than any other motion consistent with the same normal motion of the boundary. Let T be the kinetic energy of the irrotational motion to which the velocity-potential <f> refers, and Tj that of another motion given by ^"~ax ^^' ^^"■^■^^«' ^=-^ + ^0' (®^ ♦ (W. Thomaon) "On the Vis- Viva of a liquid in Motion," Camb. and Dub. Math, Joum. 1849 [Papers, t. i. p. 107]. t Thomson and Tait, Art. 312. 46 Irrotational Motion [chap, in where, in virtue of the equation of continuity, and the prescribed boundary- ' condition, we must have 3a? dy dz throughout the region, and Iuq + mvq + nw^ = over the boundary. Further let us write ^0 = i/> Hi (V + V + W7o«) dxdydz. (9) We find Ti = T + To - /> ///(^o ^^-^ ''o^ + ^o^^) dxdydz. Since the last integral vanishes, by Art. 42 (4), we have T, = T+To, (10) which proves the theorem*. 46. We shall require to know, hereafter, the form assumed by the ex- pression (4) for the kinetic energy when the fluid extends to infinity and is at rest there, being limited internally by one or more closed surfaces S. Let us suppose a large closed surface 2 described so as to enclose the whole of S. The energy of the fluid included between S and 2 is -},//*g^_j.//*i^, (11) where the integration in the first term extends over £•, that in the second over 2. Since we have, by the equation of continuity, //l^ + Z/S^-O' the expression (11) may be written -yjj(^-c)^ds-yjj{<f>-c)^^di, (12) where C may be any constant, but is here supposed to be the constant value to which <f> was shewn in Art. 39 to tend at an infinite distance from S, Now the whole region occupied by the fluid may be supposed made up of tubes of flow, each of which must pass either from one point of the internal boundary to another, or from that boundary to infinity. Hence the value of the integral // K^- taken over any surface, open or closed, finite or infinite, drawn within the region, must be finite. Hence ultimately, when 2 is taken infinitely large and infinitely distant all round from S, the second term of (12) vanishes, and we have 2T=^-pjj{<f>-C)^^dS, (13) where the integration extends over the internal boundary only. * Some eztensionB of this result are discussed by Leathern, Cambridge Tracts, No. 1« 2nd ed. (1913). They supply further interesting illustrations of Kelvin's general dynamical principle. 45-47] Cyclic Regions 47 If the total flux across the internal boundary be zero, we have so that (13) may be written 2T = - p (U ^^ dS, (14) simply. On MuUiply-connect^ Regions. 47. Before discussing the properties of irrotational motion in multiply- connected regions we must examine more in detail the nature and classification of such regions. In the following synopsis of this branch of the geometry of position we recapitulate for the sake of completeness one or two definitions already given. We consider any connected region of space, enclosed by boundaries. A region is 'connected' when it is possible to pass from any one point of it to any other by an infinity of paths, each of which lies wholly in the region. Any two such paths, or any two circuits, which can by continuous variation be made to coincide without ever passing out of the region, are said to be 'mutually reconcileable.' Any circuit which can be contracted to a point without passing out of the region is said to be 'reducible.' Two reconcileable paths, combined, form a reducible circuit. If two paths or two circuits be reconcileable, it must be possible to connect them by a continuous surface, which lies wholly within the region, and of which they form the complete boundary ; and conversely. It is further convenient to distinguish between 'simple' and 'multjiple' irreducible circuits. A 'multiple' circuit is one which can by continuous variation be made to appear, in whole or in part, as the repetition of another circuit a certain number of times. A 'simple' circuit is one with which this is not possible. A 'barrier,' or 'diaphragm,' is a surface drawn across the region, and limited by the line or lines in which it meets the boundary. Hence a barrier is necessarily a connected surface, and cannot consist of two or more detached portions. A 'simply-connected' region is one such that all paths joining any two points of it are reconcileable, or such that all circuits drawn within it are reducible. A 'doubly-connected' region is one such that two irreconcileable paths, and no more, can be drawn between any two points Ay B of it ; viz. any other path joining AB is reconcileable with one of these, or with a combination of the two taken each a certain number of times. In other words, the region is 48 Irrotational Motion [chap, hi such that one (simple) irreducible circuit can be drawn in it, whilst all other circuits are either reconcileable with this (repeated, if necessary), or are reducible. As an example of a doubly-connected region we may take that enclosed by the surface of an anchor-ring, or that external to such a ring and extending to infinity. Generally, a region such that n irreconcileable paths, and no more, can be drawn between any two points of it, or such that n — \ (simple) irreducible and irreconcileable circuits, and no more, can be drawn in it, is said to be ' w-ply-connected.' The shaded portion of the figure on p. 33 is a triply-connected space of two dimensions. It may be shewn that the above definition of an n-ply-connected space is self-consistent. In such simple cases as n = 2, n = 3, this is sufficiently evident without demonstration. 48. Let us suppose, now, that we have an n-ply-connected region, with n — \ simple independent irreducible circuits drawn in it. It is possible to draw a barrier meeting any one of these circuits in one point only, and not meeting any of the w — 2 remaining circuits. A barrier drawn in this manner does not destroy the continuity of the region, for the interrupted circuit remains as a path leading round from one side to the other. The order of connection of the region is however diminished by unity; for every circuit drawn in the modified region must be reconcileable with one or more of the w — 2 circuits not met by the barrier. A second barrier, drawn in the same manner, will reduce the order of connection again by one, and so on ; so that by drawing w — 1 barriers we can reduce the region to a simply-connected one. A simply-connected region is divided by a barrier into two separate parts; for otherwise it would be possible to pass from a point on one side the barrier to an adjacent point on the other side by a path Ipng wholly within the region, which path would in the original region form an irreducible circuit. Hence in an n-ply-connected region it is possible to draw n — 1 barriers, and no more, without destropng the continuity of the region. This property is sometimes adopted as the definition of an n-ply-connected space. Irrotational Motion tn MuUiply-conneded Spaces. 49. The circulation is the same in any two reconcileable circuits ABCA, A'B'C'A' drawn in a region occupied by fluid moving irrotationally. For the two circuits may be connected by a continuous surface lying wholly within the region; and if we apply the theorem of Art. 82 to this surface, we 47-50] Cydic Vdoeity-Potentials 49 have, remembering the rule as to the direction of integration round the boundary, I (ABC A) + / (A'CWA') = 0, or I (ABC A) = I (A'WC'A'). If a circuit ABC A be reconcileable with two or more circuits A'B'C'A\ A"B"C"A'\ &c., combined, we can connect all these circuits by a continuous surface which lies wholly within the region, and of which they form the com- plete boundary. Hence / (ABC A) + I (A'C'B'A') + I (A"C''B''A") + &c. = 0, or I (ABC A) = I(A'B'C'A') + / (A"B'VA'') + &c. ; i.e. the circulation in any circuit is equal to the sum of the circulations in the several members of any set of circuits with which it is reconcileable. Let the order of connection of the region be n + 1, so that n independent simple irreducible circuits ^i , ag , ... a„ can be drawn in it ; and let the circu- lations in these be ic ^ , ktj , ... #f „ , respectively. The sign of any k will of course depend on the direction of integration round the corresponding circuit; let the direction in which k is estimated be called the positive direction in the circuit. The value of the circulation in any other circuit can now be found at once. For the given circuit is necessarily reconcileable with some com- bination of the circuits ai^a^, ... a„; say with a^ taken j>^ times, a, taken Pj times and so on, where of course any p is negative when the corre- sponding circuit is taken in the negative direction. The required circulation then is Pl*^! + P^t + . . . + P«^n (1) Since any two paths joining two points A, B of the region together form a circuit, it follows that the values of the flow in the two paths differ by a quantity of the form (1), where, of course, in particular cases some or all of the j7's may be zero. 50. Let us denote by — ^ the flow to a variable point P from a fixed point -4, viz. <^ = — I (udx -I- vdy + wdz) (2) So long as the path of integration from ^ to P is not specified, <^ is indeter- minate to the extent of a quantity of the form (1). If however n barriers be drawn in the manner explained in Art. 48, so as to reduce the region to a simply-connected one, and if the path of integration in (2) be restricted to lie within the region as thus modified (Le. it is not to cross any of the barriers), then <^ becomes a single-valued function, as in Art. 35. It is continuous throughout the modified region, but its values at two adjacent points on opposite sides of a barrier differ by ± k. To derive the value of <f> when the integration is taken along any path in the unmodified region we must subtract the quantity (1), where any p denotes the number of L. H. 4 60 Irrotational Motion [chap, hi times this path crosses the corresponding barrier. A crossing in the positive direction of the circuits interrupted by the barrier is here counted as positive, a crossing in the opposite direction as negative. By displacing P through an infinitely short space parallel to each co-ordinate axis in succession, we find 3<i dfj> dd) SO that <f> satisfies the definition of a velocity-potential (Art. 17). It is now however a many- valued or cycUc function ; i.e. it is not possible to assign to every point of the original region a unique and definite value of <f>, such values forming a continuous system. On the contrary, whenever P describes an irreducible circuit, <f> will not, in general, return to its original value, but will differ from it by a quantity of the form (1). The quantities ici, icj* • • • '^m which specify the amounts by which <f> decreases as P describes the several independent circuits of the region, may be called the *cycUc constants' of <f>. It is an immediate consequence of the * circulation-theorem' of Art. 33 that under the conditions there presupposed the cyclic constants do not alter with the time. The necessity for these conditions is exemplified in the problem of Art. 29, where the potential of the extraneous forces is itself a cycUc function. The foregoing theory may be illustrated by the case of Art. 27 (2), where the region (as limited by the exolusion of the origin, where the formula would give an infinite velocity) is doubly-connected ; since we can connect any two points ^, B of it by two irreconcileable X>aths passing on opposite sides of the axis of z, e.g. ACBt ADB in the figure. The portion of the plane zx for which x is positive may be taken as a barrier, and the region is thus made simply-connected. The circulation in any circuit meeting this barrier once only, e,g. in ACBDA, is fi/r .rdS, or 2iru,. That in any circuit not meeting the barrier is zero. In the modified region (f) may be put equal to a single- valued function, viz. -ftB, but its value on the positive side of the barrier is zero, that at an adjacent point on the negative side is - 2rrfx. More complex illustrations of irrotational motion in multiply-connected spaces of two dimensions will present themselves in the next chapter. 51. Before proceeding further we may briefly indicate a somewhat different method of presenting the above theory. Starting from the existence of a velocity-potential as the characteristic of the class of motions which we propose to study, and adopting the second definition of an w -f 1 -ply-connected region, indicated in Art. 48, we remark that in a simply-connected region every equipotential surface must either be a closed surface, or else form a barrier dividing the region into two separate J i 60-52 J Multiple Connectivity 51 parts. Hence, supposing the whole system of such surfaces drawn, we see that if a closed curve cross any given equipotential surface once it must cross it again, and in the opposite direction. Hence, corresponding to any element of the curve, included between two consecutive equipotential surfaces, we have a second element such that the flow along it, being equal to the difference between the corresponding values of ^, is equal and opposite to that along the former ; so that the circulation in the whole circuit is zero. If however the region be multiply-connected, an equipotential surface may form a barrier without dividing it into two separate parts. Let as many such surfaces be drawn as is possible without destroying the continuity of the region. The number of these cannot, by definition, be greater than n. Every other equipotential surface which is not closed will be reconcileable (in an obvious sense) with one or more of these barriers. A curve drawn from one side of a barrier round to the other, without meeting any of the remaining barriers, will cross every equipotential surface recon- cileable with the first barrier an odd number of times, and every other equipotential surface an even number of times. Hence the circulation in the circuit thus formed will not vanish, and <f> will be a cyclic function. In the method adopted above we have based the whole theory on the equations dw dv ^ ^^ ^^ _ A ^^ _ ?!f _ n (^\ d^^di"^' di^di" ' di'^dy'^ ' ^ ^ and have deduced the existence and properties of the velocity-potential in the various cases as necessary consequences of these. In fact, Arts. 34, 35, and 49, 50 may be regarded as an inquiry into the nature of the solution of this system of differential equations, as depending on the character of the region through which they hold. The integration of (3), when we have, on the right-hand side, instead of ^ero, known functions of a;, y, z, will be treated in Chapter vn. 52. Proceeding now, as in Art. 36, to the particular case of an incom- pressible fluid, we remark that whether <^ be cyclic or not, its first derivatives dift/dx, d<f}/dy, d^jdz, and therefore all the higher derivatives, are essentially single-valued functions, so that <f> will still satisfy the equation of continuity VV = 0, (1) or the equivalent form II ^ dS = 0, (2) where the surface-integration extends over the whole boundary of any portion of the fluid. The theorem (a) of Art. 40, viz. that <{> must be constant throughout the interior of any region at every point of which (1) is satisfied, if it be constant over the boundary, still holds when the region is multiply-connected. For <f>, being constant over the boundary, is necessarily single- valued. 4—2 52 Irrotational Motion [chap, hi The remaining theorems of Art. 40, being based on the assumption that the stream-lines cannot form closed curves, wiU require modification. We must introduce the additional condition that the circulation is to be zero in each circuit of the region. Removing this restriction, we have the theorem that the irrotational motion of a liquid occupying an w-ply-connected region is determinate when the normal velocity at every point of the boundary is prescribed, as well as the value of the circulation in each of the n independent and irreducible circuits which can be drawn in the region. For ii <f>i, <f>2 be the (cyclic) velocity-potentials of two motions satisfying the above conditions, then <f> = <f>i — (f>2 is Sk single-valued function which satisfies (1) at every point of the region, and makes d<f>/dn = at every point of the boundary. Hence, by Art. 40, <f> is constant, and the motions determined by <t>i and (f>2 are identical. The theory of multiple connectivity seems to have been first developed by Biemann* for spaces of two dimensions, d propas of his researches on the theory of functions of a complex variable, in which connection also cyclic functions, satisfying the equation through multiply-connected regions, present themselves. The bearing of the theory on Hydrodynamics and the existence in certain cases of many- valued velocity-potentials were first pointed out by von Helmholtzf. The subject of cyclic irrotational motion in multiply-connected regions was afterwards taken up and fully investigated by Lord Kelvin in the paper on vortex-motion already referred to;]: . Kelvin's Extension of Green's Theorem. 53. It was assumed in the proof of Green's theorem that <f> and <{>' were both single-valued functions. If either be a cyclic function, as may be the case when the region to which the integrations in Art. 43 refer is multiply- connected, the statement of the theorem must be modified. Let us suppose, for instance, that <f> is cyclic ; the surface-integral on the left-hand side of Art. 43 (5), and the second volume-integral on the right-hand side, are then indeterminate, on account of the indeterminateness in the value of <f> itself. To remove this indeterminateness, let the barriers necessary to reduce the region to a simply-connected one be drawn, as explained in Art. 48. We may now suppose ^ to be continuous and single-valued throughout the * Orundlcigen fUr eine aUgemeine Theorie der Functumen einer verdnderlichen complexen Grosse, Qottingen, 1861 [Maihematiache Werke, Leipzig, 1876, p. 3]. Also: "Lehrs&tze aua der Analysia Situs," CreUe, t. liv. (1867) [Werke, p. 84], t CreUe, U Iv. (1868). X See also Kirchhoff, "Ueber die Krafte welche zwei unendlich dunne starre Ringe in einer fliissigkeit scheinbar auf einander ausiiben konnen," Creile, t. Ixxi. (1869) [Oea. Ahh, p. 404]. 52-54] Extenswii of Ghreen's Theorem 53 region thus modified; and the equation referred to will then hold, provided the two sides of each barrier be reckoned as part of the boundary of the region, and therefore included in the surface-integral on the left-hand side. liCt ha-i be an element of one of the barriers, k^ the cyclic constant corre- sponding to that barrier, d<f}'/dn the rate of variation of <f>^ in the positive direction of the normal to Sa^. Since, in the parts of the surface-integral due to the two sides of Sci, d<f}'/dn is to be taken with opposite signs, whilst the value of <f> on the positive side exceeds that on .the negative side by Ki, we get finally for the element of the integral due to Sa^, the value Kid<f>'ldn\ ScTi. Hence Art. 43 (5) becomes, in the altered circumstances, where the surface-integrations indicated on the left-hand side extend, the first over the original boundary of the region only, and the rest over the several barriers. The coefficient of any k is evidently minus the total flux across the corresponding barrier, in a motion of which <(>' is the velocity- potential. The values of ^ in the first and last terms of the equation are to be assigned in the manner indicated in Art. 50. If <f>' also be a cyclic function, having the cyclic constants ici', k^\ &c., then Art. 43 (6) becomes in the same way //*'^^+-7/s*'.+«'7/s! CK7o -}"••• Equations (1) and (2) together constitute Lord Kelvin's extension of Green's theorem. 54. If ^, <f}' are both velocity-potentials of a liquid, we have V^ = 0, V^' = 0, (3) and therefore / 1 (f> ^- dS + Kijl -^ dcxi + ic , / 1 ^ (^2 + • ■ • = //f|«^ + .,'//gd.,+V//g«^.+ . (4) To obtain a physical interpretation of this theorem it is necessary to explain in the first place a method, imagined by Lord Kelvin, of generating any given cyclic irrotational motion of a hquid in a multiply-connected space. 54 Irrotational Motion [chap, iri Let us suppose the fluid to be enclosed in a perfectly smooth and flexible membrane occupying the position of the boundary. Further, • let n barriers be drawn, as in Art. 48, so as to convert the region into a simply-connected one, and let their places be occupied by similar membranes, infinitely thin^ and destitute of inertia. The fluid being initially at rest, let each element of the first-mentioned membrane be suddenly moved inwards with the given (positive or negative) normal velocity —d<f>/dn, whilst uniform impulsive pressures K^p, ic^p, . . . K^p are simultaneously applied to the negative sides of the respective barrier-membranes. The motion generated will be characterized by the following properties. It will be irrotational, being generated from rest ; the normal velocity at every point of the original boundary will have the prescribed value; the values of the impulsive pressure at two adjacent points on opposite sides of a membrane will differ by the corresponding value of «:/>, and the values of the velocity-potential will therefore differ by the corresponding value of k; finally, the motion on one side of a barrier will be continuoys with that on the other. To prove the last statement we remark, first, that the velocities normal to the barrier at two adjacent points on opposite sides of it are the same, being each equal to the normal velocity of the adjacent portion of the membrane. Again, if P, Q be two consecutive points on a barrier, and if the corresponding values of ^ be on the positive side <ffp, <f>Q, and on the negative side <f>^p, ^'q, we have and therefore <f}Q — (f>p ^^<f>Q'—<f>p, i.e., if PQ = 8s, mds = d<f>'ld8. Hence the tangential velocities at two adjacent points on opposite sides of the barrier also agree. If then we suppose the barrier-menxbranes to be hquefied immediately after the impulse, we obtain the irrotational motion in question. The physical interpretation of (4), when multiplied by — />, now follows as in Art. 44. The values of pK are additional components of momentum, and those of — H d<f>/dn . da, the fluxes through the various apertures of the region, are the corresponding generalized velocities. 55. If in (2) we put <l>'=<f>, and suppose ^ to be the velocity-potential of an incompressible fluid, we find The last member of this formula has a simple interpretation in terms of the artificial method of generating cyclic irrotational motion just explained. The 54-56] Kinetic Energy 65 first term has already been recognized as equal to twice the work done by ^ the impulsive pressure (yf> applied to every part of the original boundary of the fluid. Again, pK^ is the impulsive pressure applied, in the positive direction, to the infinitely thin massless membrane by which the place of the first barrier was supposed to be occupied ; so that the expression -»// p,c,.g(fo, denotes the work done by the impulsive forces applied to that membrane; and so on. Hence (5) expresses the fact that the energy of the motion is equal to the work done by tha whole system of impulsive forces by which we may suppose it generated. In applying (5) to the case where the fluid extends to infinity and is at rest there, we may replace the first term of the third member by -pjj(4-0^dS, (6) where the integration extends over the internal boundary only. The proof is the same as in Art. 46. When the total flux across this boundary is zero, this reduces to -<.//* dS (7) The minimum theorem of Lord Kelvin, given in Art. 45, may now be extended as follows: The irrotational motion of a liquid in a multiply-connected region has less kinetic energy than any other motion consistent with the same normal motion of the boundary and the same value of the total flux through each of the several independent channels of the region. The proof is left to the reader. Sources and Sinks. 56. The analogy with the theories of Electrostatics, the Steady Flow of Heat, &c., may be carried further by means of the conception of sources and sinks. A 'simple source' is a point from which fluid is imagined to flow out uniformly in all directions. If the total flux outwards across a small closed surface surrounding the point be m, then m is called the 'strength* of the source. A negative source is called a 'sink.' The continued existence of a source or a sink would postulate of course a continual creation or annihi* lation of fluid at the point in question. 56 IrrotcUional Motion [chap, m The velocity-potential at any point P, due to a simple source, in a liquid at rest at infinity, is ff>=^ml4fnr, (1) where r denotes the distance of P from the source. For this gives a radial flow from the point, and if 88 y ^r^rOy be an element of a spherical surface having its centre at the source, we have a constant, so that the equation of continuity is satisfied, and the flux outwards has the value appropriate to the strength of the source. A combination of two equal and opposite sources ± m', at a distance hs apart, where, in the limit, 8s is taken to be infinitely small, and m' infinitely great, but so that the product m'88 is finite and equal to fi (say), is called a 'double source' of strength /x, and the line 8^, considered as drawn in the direction from — m' to + m\ is called its axis. To find the velocity-potential at any point (a?, y, z) due to a double source /x situate at {x', y', z'), and having its aids in the direction ({, m, n), we remark that, / being any continuous function, fix' + 188, y' + m88y z' + nS«) -/(a?', y\ z') ultimately. Hence, putting/ (x', y', %') = m'/4Mr, where r = {(a; - x')* + (y - y'Y + (z- «')«}*. wefind ^^.^{i^^, + m^ + nl-)\. (2) _ /x cos^ where, in the latter form, ^ denotes the angle which the line r, considered as drawn from (x', y\ z') to (x, y, z), makes with the axis (Z, m, w). We might proceed, in a similar manner (see Art. 82), to build up sources of higher degrees of complexity, but the above is sufficient for our immediate purpose. Finally, we may imagine simple or double sources, instead of existing at isolated points, to be distributed continuously over lines, surfaces, or volumes. 57. We can now prove that any continuous acyclic irrotational motion of a liquid mass may be regarded as due to a distribution of simple and double sources over the boundary. S- (4) 56-57] Sources and Sinks 57 This dependfl on the theorem, proved in Art. 44, that if ^, ^' be any two single- valued functions which satisfy V^ » 0, V^' « throughout a given region, then //*^^-//*'s«»- <») where the integration exteads over the whole boundary. In the present application, we take (^ to be the velocity-potential of the motion in question, and put <l>' = 1/r, the reciprocal of the distance of any point of the fluid from a fixed point P. We will first suppose that P is in the space occupied by the fluid. Since <f)' then becomes infinite at P, it is necessary to exclude this point from the region to which the formula (5) applies; this may be done by describing a small spherical surface about P as centre. If we now suppose SX to refer to this surface, and 8S to the original boimdary, the formula gives At the surface S we have d/dn (1/r) = — l/r*; hence if we put SS = rHuj, and finally make r = 0, the first integral on the left-hand becomes = — 4mf>p, where <^p denotes the value of <f> at P, whilst the first integral on the right vanishes. Hence ^^-iim'^-iihm^- <" This gives the value of (f> at any point P of the fluid in terms of the values of <f> and 9^/dn at the boundary. Comparing with the formulae (1) and (2) we see that the first term is the velocity-potential due to a surface distribution of simple sources, with a density — d(f>/dn per unit area, whilst the second term is the velocity-potential of a distribution of double sources, with axes normal to the surface, the density being </>, It will appear from equation (10), below, that this is only one out of an infinite number of surface- distributions which will give the same value of <f> throughout the interior. When the fluid extends to infinity in every direction and is at rest there, the surface-integrals in (7) may, on a certain understanding, be taken to refer to the internal boundary alone. To see this, we may take as external boimdary an infinite sphere having the point P as centre. The corresponding part of the first integral in (7) vanishes, whilst that of the second is equal to (7, the constant value to which, as we have seen in Art. 41, (f> tends at infinity. It is convenient, for facility of statement, to suppose (7=0; this is legitimate since we may alwaj^ add an arbitrary constant to <f>. When the point P is external to the surface, ^' is finite throughout the original region, and the formula (5) gives at once o'-iiim^-iihm'^ '" , 58 Irrotational Motion [chap, ni where, again, in the case of a liquid extending to infinity, and at rest there, the terms due to the infinitely distant part of the boundary may be omitted. 58. The distribution expressed by (7) can, further, be replaced by one of simple sources only, or of double sources only, over the boundary. Let ^ be the velocity-potential of the fluid occupying a certain region, and let^' now denote the velocity-potential of any possible acycUc irrotational motion through the rest of infinite space, with the condition that^, or^', as the case may be, vanishes at infinity. Then, if the point P be internal to the first region, and therefore external to the second, we have where 8n, 8n' denote elements of the normal to <iS, drawn inwards to the first and second regions respectively, so that djdn' = — 3/3n. By addition, we have The function (j>' will be determined by the surface- values of <^' or d(f>'/dn\ which are as yet at our disposal. Let us in the first place make </>' = </> at the surface. The tangential velocities on the two sides of the boundary are then continuous, but the normal velocities are discontinuous. To assist the ideas, we may imagine a liquid to fill infinite space, and to be divided into two portions by an infinitely thin vacuous sheet within which an impulsive pressure p<f> is applied, so as to generate the given motion from rest. The last term of (10) disappears, so that ^'-iiim-m^- ■■•■••<") that is, the motion (on either side) is that due to a surface-distribution of simple sources, of density Secondly, we may suppose that d(f>'/dn = d<f>/dn over the boundary. This gives continuous normal velocity, but discontinuous tangential velocity, over the original botmdary. The motion may in this case be imagined to be generated by giving the prescribed normal velocity — dif>/dn to every point of an infinitely thin membrane coincident in position with the boimdary. The first term of (10) now vanishes, and we have ^.=^//(^-^')|,©^. (12) * This inTestigation was first given by Green, from the point of view of Electrostatios, {.c. ante p. 44. 57-58] Stirfdce'Distrihviioiis 59 shewing that the motion on either side may be conceived as due to a surface- distribution of double sources, with density It may be shewn that the above representations of <j> in terms of simple sources alone, or of double sources alone, are unique; whereas the repre- sentation of Art. 57 is indeterminate*. It is obvious that cyclic irrotational motion of a liquid c€uinot be reproduced by any arrangement of simple sources. It is easily seen, however, that it may be represented by a certain distribution of double sources over the boundary, together with a uniform distri- bution of double sources over each of the barriers necessary to render the region occupied by the fluid simply-connected. In fact, with the same notation as in Art. 53, we find where is the single- valued velocity-potential which obtains in the modified region, and <f/ is the velocity-potential of the acyclic motion which is generated in the external space when the proper normal velocity -d<f>/dn is given to each element ^S of a membrane coincident in position with the original boundary. Another mode of representing the irrotational motion of a liquid, whether cyclic or not, will present itself in the chapter on Vortex Motion. We here close this account of the theory of irrotational motion. The mathematical reader will doubtless have noticed the absence of some im- portant links in the chain of our propositions. For example, apart from physical considerations, no proof has been offered that a function </> exists which satisfies the conditions of Art. 36 throughout any given simply- connected region, and has arbitrarily prescribed values over the boundary. The formal proof of 'existence- theorems' of this kind is not attempted in the present treatise. For a review of the literature of this part of the subject the reader may consult the authors cited below f. * Cf. Larmor, "On the Mathematical Expression of the Principle of Huyghens," Proc, Lond^ Maih. Soc, (2) t. i. p. 1 (1903). t H. Burkhardt and W. F. Meyer, "Potentialtheorie,*' and A. Sommerfeld, *'Ilandwerth- aufgaben in der Theorie d. part. Diff.-Gleiohungen," ^tieyc. d. nuUh, Wisa, t. ii. (1900). CHAPTER IV MOTION OF A LIQUID IN TWO DIMENSIONS • • • 59. If the velocities u, v be functions of Xy y only, whilst w is zero, the motion takes place in a series of planes parallel to xy^ and is the same in each of these planes. The investigation of the motion of a liquid under these circumstances is characterized by certain analjrtical peculiarities; and the solutions of several problems of great interest are readily obtained. Since the whole motion is known when we know that in the plane z = 0, we may -confine our attention to that plane. When we speak of points and lines drawn in it, we shall understand them to represent respectively the straight lines parallel to the axis of 2, and the cylindrical surfaces having their generating lines parallel to the axis of z, of which they are the traces. By the flux across any curve we shall understand the volume of fluid which in unit time crosses that portion of the cylindrical surface, having the curve as base, which is included between the planes 2; = 0, 2 = 1. Let Ay P be any two points in the plane xy. The flux across any two lines joining AP is the same, provided they can be reconciled without passing out of the region occupied by the moving liquid; for otherwise the space included between these two Unes would be gaining or losing matter. Hence if A be fixed, and P variable, the flux across any line AP is a function of the position of P. Let ^ be this function ; more precisely, let ^ denote the flux across AP from right to lefty as regards an observer placed on the curve, and looking along it from A in the direction of P. Analytically, if Z, m be the direction-cosines of the normal (drawn to the left) to any element S^ of the curve, we have 0=1 {lu-\- mv) ds (1) If the region occupied by the liquid be aperiphractic (see p. 38), is necessarily a single- valued function, but in periphractic regions the value of ^ may depend on the nature of the path AP, For spaces of two dimensions, however, periphraxy and multiple-connectivity become the same thiQg, so that 59] Stream-Function 61 the properties of 0, when it is a many-valued function, in relation to the nature of the region occupied by the moving Uquid, may be inferred from Art. 50, where we have discussed the same question with regard to ^. The cycUc constants of ^, when the region is periphractic, are the values of the flux across the closed curves forming the several parts of the internal boundary. A change, say from A U> B, oi the point from which ^ is reckoned has merely the effect of adding a constant, viz. the flux across a line BAy to the value of iji ; so that we may, if we please, regard as indeterminate to the extent of an additive constant. If P move about in such a manner that the value of ^ does not alter, it will trace out a curve such that no fluid anywhere crosses it, i.e. a stream-Une. Hence the curves ^ = const, are the stream-lines, and ^ is called the 'stream- function.' If P receive an infinitesimal displacement PQ (= hy) parallel to y, the increment of ^ is the flux across PQ from right to left, i.e. Bt/t = — u . PQ^ or «=-^^ -(2) Again, displacing P parallel to x, we find in the same way «-g <») The existence of a function iff related to u and v in this manner might also have been inferred from the form which the equation of continuity takes in this case, viz. ai + a^ = ^' (*) which is the analjrtical condition that udy — vdx should be an exact difEerential*. The foregoing considerations apply whether the motion be rotational or irrotational. The formulae for the components of vorticity, given in Art. 30, become (-0. ,-«, J-S + l^; (5) SO that in irrotational motion we have a^2+ap = o (^) * The function ^ was introdnced in this way by Lagrange, Nouv. nUm. de VAcad. de Berlin, 1781 [Oeuvres, t. iv. p. 720]. The kinematical interpretation is due to Rankine, "On Plane Water-lines in Two Dimensions," PhU. Trans, 1864 [MiaceOaneotu Scientific Papersj London, 1881, p. 496]. 62 Motion of a Liquid in Two Dimensions [chap, iv 60. In what follows we confine ourselves to the case of irrotational motion, which is, as we have already seen, characterized by the existence, in addition, of a velocity-potential <^, connected with w, v by the relations «-!• »--| <" and, since we are considering the motion of incompressible fluids only, satisfying the equation of continuity aii + a^-" ^^' The theory of the function <^, and the relation between its properties and the nature of the two-dimensional space through which the irrotational motion holds, may be readily inferred from the corresponding theorems in three dimensions proved in the last chapter. The alterations, whether of enunciation or of proof, which are requisite to adapt these to the case of two dimensions are for the most part purely verbal. An exception, which we will briefly examine, occurs howerer in the case of the theorem of Art. 39 and of those which depend upon it. If dd be an element of the boundary of any portion of the plane xy which is occupied wholly' by moving liquid, and if dn be an element of the normal to ha drawn inwards, we have, by Art. 36, I Sn*-* (») the integration extending roimd the whole boundary. If this boimdary be a circle, and if r, ^ be polar co-ordinates referred to the centre P of this circle as origin, the last equation may be written I ^.rrfd=0, or g- I <l>de=0. 1 /■««' Hence the integral ^ j (l>d$. i.e. the mean value of </> over a circle of centre P and radius r, is independent of the value of r, and therefore remains unaltered when r is diminished without Umit, in which case it becomes the value of <f) at P. If the region occupied by the fluid be periphractio, and if we apply (3) to the space enclosed between one of the internal boundaries and a circle with centre P and radius r surrounding this boundary, and lying wholly in the fluid, we have where the integration in the flrst member extends over the circle only, and M denotes the flux into the region across the internal boundary. Hence M i: ""^.rde^-Mi (4) ^'kiy^^^' • 1 fi^ M which gives on integration ^ \ ^(W = -5- log r + 0; (5) i.e. the mean value of <f> over a circle with centre P and radius r is equal to - if /2tr . log f-¥Cy 60-61J Khietic Energy 63 where C is independent of r but may vary with the position of P. This formula holds of course only so far as the circle embraces the same internal boundary, and lies itself wholly in the fluid. If the region be unlimited eztemaUy, and if the circle embrace the whole of the internal boundaries, and if further the velocity be everywhere zero at infinity, then C is an absolute constant; as is seen by reasoning similar to that of Art. 41. It may then be shewn that the value of at a very great distance r from the internal boundary tends to the value - MI2rr . log r + 0. In the particular case of j9f =0 the limit to which tends at infinity is finite; in all other cases it is infinite, and of the opposite sign to M. We infer, as before, that there is only one single- valued function <f> which satisfies the equation (2) at every point of the plane xy external to a given system of closed curves, makes the value of d<^/3n equal to an arbitrarily given quantity at every point of these curves, and has its first differential coefficients all zero at infinity. If we imagine point-sources, of the type explained in Art. 56, to be distributed uni- formly along the axis of 2, it is readily found that the velocity at a distance r from this axis will be in the direction of r, and equal to m/2rrr, where m is a certain constant. This arrangement constitutes what may be called a * line-source ,* and its velocity-potential may be taken to be *=-^log»' -(6) The reader who is interested in the matter will have no difficulty in working out a theory of two-dimensional sources and sinks, similar to that of Arts. 56 — 58*. 61. The kinetic energy T of a portion of fluid bounded by a cylindrical surface whose generating lines are parallel to the axis of z, and by two planes perpendicular to the axis of z at unit distance apart, is given by the formula ^^-'//{(i)"-(^)>*--/*l* '" where the surface-integral is taken over the portion of the plane xy cut off by the cylindrical surface, and the line-integral round the boundary of this portion. Since d<f>/dn = — dift/dsy the formula (1) may be written 2r = />J^#, (2) the integration being carried in the positive direction round the boundary. If we attempt by a process similar to that of Art. 46 to calculate the energy in the case where the region extends to infinity, we find that its value is infinite, except when the total flux outwards (Jf ) is zero. For if we introduce a circle of great radius r as the external boundary of the portion of the plane xy considered, we find that the corresponding part of the integral on the right-hand side of (1) increases indefinitely with r. The only exception is when Jf =0, in which case we may suppose the line-integral in (1) to extend over the internal boundary only. If the cylindrical part of the boundary consist of two or more separate portions one of which embraces all the rest, the enclosed region * This subject has been treated very fully by C. Neumann, UAer das logarithmisehe und NewUm'ache Potential, Leipsig, 1877. 64 Motion of a Liquid in Two Dimensioiis [chap, iv is multiply-connected, and the equation (1) needs a correction, which may be applied exactly as in Art. 55. 62. The functions <^ and ^ are connected by the relations d<f> __d^ d<f> ^ dtff . dx~^ dy* dy ~~ dx ^ ^ These conditions are fulfilled by equating ^ + 10, where i stands as usual for \/( — 1), to any ordinary algebraic or transcendental function of x + iy, say <t> + i^=f{x-hiy) (2) For then |^ (<^ + i^) = if {x + iy) = *" ^ (^ + **0), (3) whence, equating separately the real and the imaginary parts, we see that the equations (1) are satisfied. Hence any assumption of the form (2) gives a possible case of irrotational motion. The curves </> = const, are the curves of equal velocity^potential, and the curves ^ = const, are the stream-lines. Since, by (1), d<f> dtjt d<f>d^ ^ ^ dx dx dy dy ' we see that these two systems of curves cut one another at right angles, as already proved. Since the relations (1) are unaltered when we write — \ft for <f>, and (f> for 0, we may, if we choose, look upon the curves ^ = const, as the equipo'tential curves, and the curves <l> = const, as the stream-lines ; so that every assumption of the kind indicated gives us two possible cases of irrotational motion. For shortness, we shall through the rest of this chapter follow the usual notation of the Theory of Functions, and write z^x-^-iy, (4) w = <f) -{- iff/ (5) From a modem point of view, the fundamental property of a function of a complex variable is that it has a definite differential coefiicient with respect to that variable*. Tf ^, iff denote any functions whatever of x and y, then corresponding to every value of a; 4- iy there must be one or more definite values of ^ + iif/ ; but the ratio of the differential of this function to that oi x + iy, viz. 8a: + iSy * hx + r 8y ' * See, for example, Forsyth, Theory of Functions^ 2nd ed., Cambridge, 1900, cc. i., ii. 61-62] Complex Variable 66 depends in general on the ratio 8x : hy. The condition that it should be the same for all values of the latter ratio is |+'|-'(S+'D ■■<'" which is equivalent to (1) above. This property was adopted by Biemann as the definition of a function of the complex variable x-^ iy\ viz. such a function must have, for every assigned value of the variable, not only a definite value or system of values, but also for each of these values a definite differential coefficient. The advantage of this definition is that it is quite independent of the existence of an analytical expression for the function. If the complex quantities z and w be represented geometrically after the manner of Argand and Gauss, the differential coefficient dwjdz may be interpreted as the operator which transforms an infinitesimal vector hz into the corresponding vector hw. It follows then, from the above property^ that corresponding figures in the planes of z and w are similar in their infinitely small parts. For instance, in the plane of w the straight lines ^ = const., ^ = const., where the constants have assigned to them a series of values in arithmetical progression, the common difference being infinitesimal and the same in each case, form two systems of straight lines at right angles, dividing the plane into infinitely small squares. Hence in the plane xy the corresponding curves ^ = const., ^ = const., the values of the constants being assigned as before, cut one another at right angles (as has already been proved otherwise) and divide the plane into infinitely small squares. Conversely, if ^, '^ be any two functions of ar, y such that the curves ^=mc, yft=n€, where c is infinitesimal, and m, n are any integers, divide the plane xy into elementary squares, it is evident geometrically that If we take the upper signs, these are the conditions that x +iy should be a function of <f) +%ylt. The case of the lower signs is reduced to this by reversing the sign of ^. Hence the equation (2) contains the complete solution of the problem of orthomorphic projection from one plane to another*. The similarity of corresponding infinitely small portions of the planes w and z breaks down at points where the differential coefficient dw/dz is zero or infinite. Since t't+'t (') the corresponding value of the velocity, in the hydrodynamical application, is zero or infinite. * Lagrange, "Sur la construction des cartes g^graphiques/* Nouv, nUm, de VAcad. de Berlin, 1779 [Oeiivres, t. iv. p. 636]. For the further history of the problem, see Forsyth, Theory of Functions, c. xix. , L. H. 5 (1) 66 Motion of a Liquid in Two Dimensions [chap, iv In all. physical applications, w must be a single- valued, or at most a cyclic function of 2, in the sense of Art. 50, throughout the region with which we are concerned. Hence in the case of a 'multiform' function, this region must be confined to a single sheet of the corresponding Biemann's surface, and * branch-points* therefore must not occur in its interior. 63. We can now proceed to some applications of the foregoing method. First Jet us assume w = Az^, A being real. Introducing polar co-ordinates r, d, we have if, = Ar^ cos ndy ' ^ = Ar^ Binnd. ) The following cases may be noticed. 1^. If n = 1, the stream-lines are a system of straight lines parallel to x, and the equipotential curves are a similar system parallel to y. In this case any corresponding figures in the planes of w and z are similar, whether they be finite or infinitesimal. 2^. If n= 2, the curves </> = const, are a system of rectangular hyperbolas having the axes of co-ordinates as their principal axes, and the curves ^ = const, are a similar system, having the co-ordinate axes as asymptotes. The lines 6 = 0, = ^tt are parts of the same stream-line ^ = 0, so that we may take the positive parts of the axes of x, y as fixed boundaries, and thus obtain the case of a fluid in motion in the angle between two perpendicular walls. 3°. If n = — 1, we get two systems of circles touching the axes of co-ordinates at the origin. Since now <f) = A/r . cos 0, the velocity at the origin is infinite ; we must therefore suppose the region to which our formulae apply to be limited internally by a closed curve. 4°. If n = — 2, each system of curves is composed of a double system of lemniscates. The axes of the system <f> = const, coincide with a; or y ; those of the system = const, bisect the angles between these axes. 5°. By properly choosing the value of w we get a case of irrotational motion in which the botmdary is composed of two rigid walls inclined at any angle a. The equation of the stream-lines. being T^BinnO = const., (2) we see that the lines = 0, = ir/n are parts of the same stream-line. Hence if we put n = irja, we obtain the required solution in the form <f) = Ar^ cos — , Jt = Ar^ sin — (3) a a 62-64] Examples 67 The component velocities along and perpendicular to r are — il-f COS — , and A-r sm — , a <x a a and are therefore zero, finite, or infinite at the origin, according as a is less than, equal to, or greater than tt. 64. We take next some cases of cyclic functions. 1°. The assumption «r = — /x log 2, (1) where /i is real, gives ^ = — ft log r, = — /xfl (2) The velocity at a distance r from the origin is /i/r ; this point must therefore be isolated by drawing a closed curve round it. If we take the radii 6 = const, as the stream-lines we get the case of a (two-dimensional) source at the origin. (See Art. 60.) If the circles r = const, be taken as stream-lines we have the case of Art. 27 ; the motion is now cycUc, the circulation in any circuit embracing the origin being 27rfi. 2°. Let us take w = — u log — ; — (3) If we denote by r^, r^ the distances of any point in the plane xy from the points (± a, 0), and by 0^, ^2 ^^^ angles which these distances make with the positive direction of the axis of a;, we have whence ^ = — ft log fi/r,, ^ = — ft (^i — B^ (4) The curves ^ = const., ^ == const, form two orthogonal sjrstems of 'coaxal' circles ; see p. 68. Either of these systems may be taken as the equipotential curves, and the other system will then form the stream-Unes. In either case the velocity at the points (± a, 0) will be infinite. If these points be accordingly isolated by drawing closed curves round them, the rest of the plane xy becomes a triply-connected region. If the circles 6^ — 62"= const, be taken as the stream-lines we have the case of a source and a sink, of equal intensities, situate at the points (± a, 0). If a is diminished indefinitely, whilst iia remains finite, we reproduce the assumption of Art. 63, 3^, which therefore corresponds to the case of a double line-source at the origin. (See the first diagram of Art. 68.) If, on the other hand, we take the circles r-^r^ = const, as the stream-lines we get a case of cyclic motion, viz. the circulation in any circuit embracing the first (only) of the above points is 27rft, that in a circuit embracing the 6—2 68 Motion of a Liquid in Two Dimensions [chap, iv second is — 27r/x ; whilst that in a circuit embracing both is zero. This example will have additional interest for us when in Chapter vii. we come to treat of 'Kectilinear Vortices.' 3°. The potential- and stream-functions due to a row of equal and equidistant sources at the points (0, 0), (0, ± a), (0, ± 2a), . . . are given by the formula u; oclog 2 4- log (z — ia) 4- log (2 4- io) 4- log {z — 2ia) 4- log (z 4- 2ia) 4- . . . , (5) ttz or, say, w = C log sinh — , (6) a where C is real. This makes in agreement with a result given by Maxwell*. The formulae apply also to the case of a source midway between two fixed boundaries y = ± ^a. The case of a row of double sources having their axes parallel to x is obtained by differentiating (6) with respect to z. Omitting a factor we have w = Ccoth — , a (8) ^j . _ C sinh (27ng/a) ,^ C sin (27Ty/g) ^ cosh {^TTx/a) — cos (27ry/a) ' ^ cosh (^ttx/o) — cos (iiTy/a) ' "^ ^ * Electricity and Magnetism, Art. 203. 64-65] Inverse Formulae 69 Superposing a uniform motion parallel to x negative, we have w? = 2 + C coth ^ , (10) a or , __ C sinh (iirxja) . _ C sin {^myja) ^ cosh {27rx/a) — cos (^Try/a) ' ^ ~ ^r ^^gj^ ^27Tx/a) — cos (^Try/a) ' (11) The stream-line ^ = now consists in part of the line y = 0, and in part of an oval curve whose semi-diameters parallel to x and y are given by the equations sinh«!!? = ![2, ytan^ = C (12) If we put C = Trb^a, (13) where 6 is small compared with a*, these semi-diameters are each equal to i, approximately. We thus obtain the potential- and stream-functions for a liquid flowing through a grating of parallel cylindrical bars of small circular section. The second of equations (11) becomes in fact, for small values of x, y^ ^=yi^-^^ (1^) x^ + y\ 65. If ti; be a function of z, it follows at once from the definition of Art. 62 that 2; is a function of w. The latter form of assumption is some- times more convenient analytically than the former. The relations (1) of Art. 62 are then replaced by dx_dy dx__dy ,,. d4>'^d^' difs" d<f> ^^ Also smce -^- = ^-1-^^ = — w + *v, , dz 1 1 /u , ,v\ we have — — - = == _ ( _ 4- ^ - dw u — IV q\q qj where q is the resultant velocity at (ar, y). Hence if we write i—t <2> and imagine the properties of the function ^ to be exhibited graphically in the manner already explained, the vector drawn from the origin to any point in the plane of t, will agree in direction with, and be in magnitude the reciprocal of, the velocity at the corresponding point of the plane of z. * The approzimately ciroular form holds however for a considerable range of values of C. Thus if we put C=ia, we find from (12) a:/a = -264, y fa = -250. The two diameters are very nearly equal, although the breadth of the oval is half the interval between the stream-lines y = ± Ja. 70 Motion of a Liquid in Two Dimensions [chap, iv Again, since 1/g is the modulus of dzjdwy i.e. of dxjd^ + idy/d<f>, we have h-&'H%)' <'> which may, by (1), be put into the equivalent forms (4) The last formula, viz. -z = .A-r-iiy (5) expresses the fact that corresponding elementary areas in the planes of z and w are in the ratio of the square of the modulus of dz/dw to unity. 66. The following examples of this procedure are important. l^ Assume z — ccoshw, (1) or a? = c cosh <f> cos ^,1 .^. y = csinh^sin 0.) The curves <f> = const, are the ellipses _^!_ + __lL__i (3) c» co8h« <f,^e* sinh* <f>~ ' ^ ' and the curves tfi = const, are the hyperbolas ^' t 1 (4) c* COS* ^ c* sin* ^ ' these conies having the common foci (± c, 0). The two systems of curves are shewn on the opposite page. Since at the foci we have <f> = 0, ff/ = nir, n being some integer, we see by (2) of the preceding Art. that the velocity there is infinite. If the hyperbolas be taken as the stream-lines, the portions of the axis of x which lie outside the points (± c, 0) may be taken as rigid boundaries. We obtain in this manner the case of a liquid flowing from one side to the other of a thin plane partition, through an aperture of breadth 2c; the velocity at the edges is however infinite. If the ellipses be taken as the stream-lines we get the case of a liquid circulating round an elliptic cylinder, or, as an extreme case, round a rigid lamina whose section is the line joining the foci {± c, 0). At an infinite distance from the origin <f} is infinite, of the order Togr, where r is the radius vector; and the velocity is infinitely small of the order 1/r. 65-66] Examples 71 2^ Let 2 = w + e^ (5) or x = <f> + e* cos ^, y = ^ + c* sin ^ (6) « The stream-line ^ = coincides with the axis of x. Again, the portion of the line y = tt between x = — oo and a; = — 1, considered as a line bent back on itself, forms the stream-line ^ = tt ; viz. as ^ decreases from + oo through to — 00 , X increases from — oo to — 1 and then decreases to — oo again. Similarly for the stream-line ^ = — tt. Since il = — dz/dw = — 1 ~ e* cos ^ — le* sin ^, it appears that for large negative values of <f> the velocity is in the direction of x-negative, and equal to unity, whilst for large positive values it is zero. The above formulae therefore express the motion of a liquid flowing into a canal bounded by two thin parallel walls from an open space. At the ends of the walls we have ^ = 0, ^ = ± tt, and therefore ? = 0, i.e. the velocity is infinite. The direction of the flow will be reversed if we change the sign of w in (5). The forms of the stream-lines, drawn, as in aU similar cases in this chapter, for equidistant values of ^, are shewn on the next page*. * This example was given by Helmholtz, Berl MonaUber. April 23, 1868 [PhU, Mag. Nov. 1868, Wise, Ahh, t. i. p. 164]. 72 Motion of a Liquid in Two Dimensions [chap, iv If the walls instead of being parallel make angles ± p with the line of symmetry, the appropriate formula is z = n (1 __ g-n») ^ g(l-n)w (7) where n = p/ir. The stream-lines ^ = ± tt follow the course of the walls*. This agrees with (5) when n tends to the limit 0, whilst if n = J we have virtually the case shewn on the preceding page. 67. It is known that a function / (z) which is finite, continuous, and single-valued, and has its first derivative finite, at all points of the space included between two concentric circles about the origin, can be expanded in the form f{z) = Ao + A^z 4- A^z^ 4- . . . + B^z-^ + B^z-^ + (1) If the above conditions be satisfied at all points within a circle having the origin as centre, we retain only the ascending series ; if at all points without such a circle, the descending series, with the addition of the constant A^^ is sufficient. If the conditions be fulfilled for all points of the plane ani without exception,/ (2) can be no other than a constant A^. * R. A. Harris, "On Two-Dimendonal fluid Motion through Spouts composed of two Plane WaUs,'* Ann. of Math, (2), t. ii. (1901). A diagram is given for the case of /3=jtx. 66-68] General Formvlae 73 Putting / (2) =^ + i^, introducing polar co-ordinates, and writing the complex constants A^ , B^ in the forms P^ 4- iQ^ , R^ + iS^ , respectively, we obtain ^ = Po + 2rr«(P„cosne-gn8inne) + 2rr-«(fi„cosne + /S„sinne),) = Qo + ^r^** (On cos n^ + P„ sin nJd) + S^f-" (iS„ cos nfi - fi„ sin wfl) J These formulae are convenient in treating problems where we have the value of <f>, or of d<f>ldn^ given over the circular boundaries. This value may be expanded for each boundary in a series of sines and cosines of multiples of 0, by Fourier's theorem. The series thus found must be equivalent to those obtained from (2) ; whence, equating separately coefficients of sin r\d and cosnd, we obtain equations to determine P„, Q„, 22^, iS^. 68. As a simple example let us take the case o f an infinitely long circular y^/'- ^^^ cylinder of radius a moving with velocity U perpendicular to its length, in an ^ infinite mass of liquid which is at rest at infinity. Let the origin be taken in the axis of the cylinder, and the axes of x, y in a plane perpendicular to its length. Further let the axis of x be in the direction of the velocity t7. The motion, supposed originated from rest, will necessarily be irrotational, and <f> will be single- valued. Also, since jd<f>/dn . ds, taken round the section of the cylinder, is zero, ^ is also single-valued •(Art. 59), so that the formulae (2) apply. Moreover, since 3^/9n is given at every point of the internal boundary of the fluid, viz. - ^ = t7 cos «, f or r = a, (3) and since the fluid is at rest at infinity, the problem is determinate, by Art. 41. These conditions give P„ = 0, Qn = 0, and U cos ^ = 2" na-^'^ (22„ cos nB -h S^sm nd), which only c^n be satisfied by making Ri = Ua\ and all the other coefficients zero. The complete solution is therefore <f> = — cos 6, ^ = BinO (4) The stream-Unes ^ = const, are circles, as shewn on the next page. The kinetic energy of the liquid is given by the formula (2) of Art. 61, viz. 2T = pUd^ = pU^^ j^coB^ede = M'U\ (5) if ibf ', = ^na^p, be the mass of fluid displaced by unit length of the cylinder. This result shews that the whole effect of the presence of the fluid may b e represe nted by an addition M' to the inertia per umt le ngth of the cyhnde r. 74 Motion of a Liquid in Two Dimendons [chap, iy Thus, in the case of lectilinear motion, if we have an extraneous force X per unit length acting on the cylindei, the equation of energy gives or {M + M')^ = X, dt (6) where M represents the mass of the cylinder itself. Writing this in the form dt dt' we learn that the pressure of the fluid is equivalent to a force — M'dUjdt per unit length in the direction of motion. This vanishes when TJ is constant. The above result must of course admit of verification by direct calculation. By Art. 20 (6) the pressure is given by the formula M-fe'-^c) (') provided q denote the velocity of the fluid relative to the centre of the moving sphere. The term due to the extraneous forces (if any) acting on the fluid has been omitted ; the effect of these would be given by the rules of Hydrostatics. We have, for r =a, ^ = a^cosd, g«=4C^«sin«d, (8) whence p =p (a -^- cos 6 -2U^ sin« 6 + J^ (O) (9) 68-69] Motion of a Circular Cylinder 75 The resultant fcxrce on unit length of the cylinder is evidently parallel to the initial line ^=0; to find its amount we multiply by -adB , cos B and integrate with respect to 6 between the limits and 2v, The result is -M'dU/dt^ as before. If in the above example we impress on the fluid and the cylinder a velocity — t7 we have the case of a current flowing with the general velocity U past a fixed cylindrical obstacle. Adding to <f> and tft the terms Ur cos and Ur sin 0, respectively, we get <f>=u(r + ^cod0, ^^U(r-^)sm0. (10) If no extraneous forces act, and if {7 be constant, the resultant force on the cylinder is zero. Cf. Art. 92. 69. To render the formula (1) of Art. 67 capable of representing any case of continuous irrotational motion in the space between two concentric circles, we must add to the right-hand side the term ^log^ (1) li A ^ P -i- iQ, the corresponding terms in ^, ^ are P]osr^Q0, P0-{-Qlogr ......(2) respectively. The meaning of these terms is evident; thus 2nPy the cycUc constant of iff, is the flux across the inner (or outer) circle; and 27tQ, the cyclic constant of <f>, is the circulation in any circuit embracing the origin. For example, returning to the problem of the last Art., let us suppose that in addition to the motion produced by the cylinder we have an independent 76 Motion of a Liquid in Two Dimensions [chap, iv circulation round it, the cyclic constant being k. The boundary-condition is then satisfied by a' K ^ r 27T (3) The effect of the cyclic motion, superposed on that due to the cylinder, will be to augment the velocity on one side, and to diminish (and, it may be, to reverse) it on the other. Hence when the cylinder moves in a straight line with constant velocity, there will be a diminished pressure on one side, and an increased pressure on the other, so that a constraining force must be applied at right angles to the direction of motion. The figure shows the lines of flow. At a distance from the origin they approximate to the form of concentric circles, the disturbance due to the cylinder becoming small in com- parison with the cyclic motion. When, as in the case represented, U > ic/2ira, there is a point of zero velocity in the fluid. The stream-line system has the same configuration in all cases, the only effect of a change in the value of U being to alter the scale, relative to the diameter of the cylinder. When the problem is reduced to one of steady motion we have in place of (3) <f> = v(r.^) COS 6 -^r-B, (4) whence -= const. -\if P = const •-l(: 2Usine^.^y (5) 69] Cylinder with Circulation 77 /: for r=a. The resultant pressure on the cylinder is therefore Sir I? sin Sad6 = -KpU, (6) at right angles to the general direction of the stream. This result is independent of the radius of the cylinder. It is not difficult indeed to shew, with the help of principles developed later in this treatise, that it holds for any form of section*. To calculate the effect of the fluid pressures on the cylinder when moving in any manner we may conveniently adopt moving axes, the origin being taken at the centre, and the axis of x in the direction of the velocity U. If ;( be the angle which this makes with a fixed direction, the equation (6) of Art. 20 gives p~ dt ^ dt de' i^'-tfe (^) where q now denotes fluid velocity relative to the origin, to be calculated from the relative velocity-potential <f> + Ur cos 3, <t> being given by (3). We find, for r=a, ? = «fcoe^-i(2t;sin^+4)%aC;|Bin^+JLj («) The resultant pressures parallel to x and y are therefore - (^'pcoBBad0='M'^, - l^' pHinBadO^^KpU -M'U^ (9) Jo at J <U where M' = irpa* as before. Hence if P, Q denote the components of the extraneous forces, if any, acting on the cylinder in the directions of the tangent and the normal to the path, respectively, the equations of motion of the cylinder are \ (10) If there be no extraneous forces, U is constant, and writing dxidi = VIR, where R is the radius of curvature of the path, we find R = (M+M') U/kp (11) The path is therefore a circle, described in the direction of the cyclic motion f. liifTfhe the rectangular co-ordinates of the axis of the cylinder, the equations ( 10) aro equivalent to {M+M')ij= kp( + yJ where X, Y are the components of the extraneous forces. To find the effect of a constant force, we may put X={M+M')g\ y=0 (13) The solution then is f = a + c cos (ni + c), (14) ri=p + - <+csin(n«+€), where a, /3, c, r are arbitrary constants, and n = Kpl(M +3f' ) (15) * This remark is due to Kutta and Joukowaki; see Kutta, Sitzb. d. k, hayr, Akad. d, Wias. 1910. t Rayleigh, "On the Irregular Flight of a Tennis Ball," Mesa, of Math. t. vii. (187S) [Papers, t. i. p. 344]; Greenhill, Mess, of Math. t. iz. p. 113 (1880). 78 Motion of a Liquid in Two Dimensions [chap, iv This shews that the path is a troohoid, deeoribed with a mean velocity g'jn perpendicular to a;*. It is remarkable that the cylinder has on the whole no progressive motion in the direction of the extraneous force. In the particular case c =0 its path is a straight line perpendicular to the force. The problem is an illustration of the theory of 'gyrostatic systems,* to be referred to in Chapter vi. 70. The formula (I) of Art. 67, as amended by the addition of the term A log z, may readily be generalized so as to apply to any case of irrotational motion in a region with circular boundaries, one of which encloses all the rest. In fact, for each internal boundary we have a series of the form 41og(«-c) + -^'^ + ^-^,+ .... where c,^a-^ib say, refers to the centre, and the coefficients A^ A^, A^, ... are in general complex quantities. The difficulty however of determining these coefficients so as to satisfy given boundary conditions is now so great as to render this method of very limited application. Indeed the determination of the irrotational motion of a liquid subject to given boundary conditions is a problem whose exact solution can be effected by direct processes in only a very few cases f. Most of the cases for which we know the solution have been obtained by an inverse process ; viz. instead of tr}ring to find a value of ^ or ^ which satisfies V\f> » or V^ = and given boundary conditions, we take some known solution of the differential equations and enquire what boundary conditions it can be made to satisfy. Examples of this method have already been given in Arts. 63, 64, and we may carry it further in the following two important cases of the general problem in two dimensions. 71. Case I. The boundary of the fluid consists of a rigid cylindrical surface which is in motion with velocity 17 in a direction perpendicular to its length. Let us take as axis of x the direction of this velocity Z7, and let S^ be an element of the section of the surface by the plane xy. Then at all points of this section the velocity of the fluid in the direction of the normal, which is denoted by ^;?^» must be equal to the velocity of * GMQhkll It. t A T«ry pow«ffal melhod of transfomiatioii, applkable to cases where the boundaries of the fluid conaisl of iixMl plane walls, has however been developed by Schwajrx, "Ueber einige Abbildiin^satt^bMU** OcUt, l« Ixx. [CawiittiMlfo J6Aaii4liiiife», Berlin. 1890» t. iL il 65]; €1iristoffel» ^^'Snl problems deQe tempeialnre stazionarie e la rapptesentaiione di una data saperfici^'* Anmsiii di M^Ummtiea (2), t. i pw S9» and Kirchhoff, '^Zor Tbeorie des Oandensalocs^" BvL Mo v ^ hUr. March 15, 1877 \iU^ Ahk. p^ 101]. Many of the aolntians which can be thus obtained are of great interr^ in the mathematieaUj cognate snbjects of Ekctrostaties, Heat- Oondiaelioa, Jbw $ee« for example, J. J. Thomson* Reemi Resemrckes • a EUctncUif «W Jfaynetfim. Oxioid. l^SO, e. iii 6&-71] Inverse Methods 79 the boundary normal to itself, or — TJdylda. Integrating along the section, we have ^= — Uy + const (1) If we take any admissible form of ^, this equation defines a system of curves each of which woidd by its motion parallel to x give rise to the stream-lines ^ = const.* We give a few examples. 1°. If we choose for iff the form — Uy^ (1) is satisfied identically for all forms of the boundary. Hence the fluid contained within a cylinder of any shape which has a motion of translation only may move as a solid body. If, further, the cyUndrical space occupied by the fluid be simply-connected, this is the only kind of motion possible. This is otherwise evident from Art. 40; for the motion of the fluid and the solid as one mass evidently satisfies all the conditions, and is therefore the only solution which the problem admits of. 2°. Let ^ = A/r . sin &; then (I) becomes — sin ^ = — Ur sin d + const (2) r ^ ' In this system of curves is included a circle of radius a, provided A/a = — Ua. Hence the motion produced in an infinite mass of liquid by a circidar cylinder moving through it with velocity u perpendicular to its length, is given by ^=-^%infl, (3) which agrees with Art. 68. 3°. Let us introduce the elliptic co-ordinates ^, rj, connected with x, y by the relation x-\' iy = c cosh (f -f iiy), (4) or x = c cosh ^ cos 97,1 .^. y = csinh ^sinij, j (cf . Art. 66), where f may be supposed to range from to 00 , and 17 from to 27r. If we now put ^ 4- i^ = Ce-(^+^''>, (6) where G is some real constant, we have ^ = - Ce-f siniy, (7) so that (1) becomes Ce~* sin 17 == t7c sinh ^ sin 17 -}- const. In this system of curves is included the ellipse whose parameter ^0 ^ determined by Ce^ = Uc sinh ^0 • * Cf. Rankine, 2.c. ante p. 61, where the method is applied to obtain curves resembling the lines of shipSi 80 Motion of a Liquid in Two Dimensions [chap, iv If a, b be the semi-axes of this ellipse we have a = c cosh ^o> b = c sinh ^q, Ubc __ jj. fa + b\\ so that C = a — b \a — 6 a 4- b\\ 7j e^^sin^y (8) gives the motion of an infinite mass of liquid produced by an elliptic cylinder of semi-axes a, 6, moving parallel to the greater axis with velocity TJ , That the above formulae make the velocity zero at infinity appears from the consideration that, when ^ is large, hx and 8y are of the same order as e^S^ or c^Sjy, so that dtff/dx, hffjdy are of the order e~^ or 1/r*, ultimately, where r denotes the distance of any point from the axis of the cylinder. If the motion of the cylinder were parallel to the minor axis, the formula would be ^=^«(— ft)* «-*<«» ^ (9) The stream-lines are in each case the same for all confocal elliptic forms of the cylinder, so that the formulae hold even when the section reduces to the straight line joining the foci. In this case (9) becomes ^ = Fc c-f cos ly, (10) 71] Translation of an Elliptic Ct/linder 81 which would give the motion produced by an infinitely long lamina of breadth 2c moving 'broadside on' in an infinite mass of liquid. Since however this solution makes the velocity infinite at the edges, it is subject to the practical Umitation already indicated in several instances*. The kinetic energy of the fluid is given by 2T = pj<f>cklf = pCh'^^* I "" C08« rfdrj =^7rph^V\ (11) where h is the half-breadth of the cylinder perpendicular to the direction of motion. If the units of length and time be properly chosen we may write X +iy =co8h (( +iff), <t> +»> =6" ^^■^•''^ whence ^ = ,^ (n.^._L_) , y=^(i__L_). These formulae are convenient for tracing the curves (^= const.. ^= const., which are figured on the preceding page. By superposition of the results (8) and (9) we obtain, for the case of an elliptic cylinder having a velocity of translation whose components are {7, F, Vr = - (^)**"^(^^ sin 17 - Fa cos ly) (12) To find the motion relative to the cylinder we must add to this the expression Uy- Fa;=c(f7 8inhf siniy- Fcosh f cos 17) ^ (13) For example, the stream-function for a current impinging at an angle of 45° on a plane lamina whose edges are at a? = ± c is V^ = -;y2 ^0^ ^^ ^ (*^^ ^ -fidni?), .(14) * This investigation was given in the Quart. Joum, of McUh, t. xiv. (1875). Results equivalent to (8), (9) had however been obtained, in a different manner, by Beltrami, "Sui prinoipii fondamentali dell' idrodinamica razionale," Mem, delV Accad, deUe Scienze di Bologna^ 1873, p. 394. L. H. 6 82 Motion of a Liquid in Two Dimensions [chap, iv where q^ is the velocity at infinity. This immediately verifies, for it makes ^ =0 for f =0, and gives V'=-||(x-y) for ^ = 00 . The stream-lines for this case djce shewn in the annexed figure (turned through 45° for convenience)*^. This will serve to illustrate some results to be obtained later in Chapter VL If we trace the course of the stream-hue -^ =0 from <^=+oo to<^=-oo,we find that it consists in the first place of the hyperbolic arc i; -\rr, meeting the lamina at right angles; it then divides into two portions, following the faces of the lamina, which finally re-unite and are continued as the hyperbolic arc i7=jir. The points where the hyperbolic arcs abut on the lamina dse points of zero velocity, and therefore of maximum pressure. It is plain that the fluid pressures on the lamina are equivalent to a couple tending to set it broadside on to the stream; and it is easily found that the moment of this couple, per unit length, is \irpq^c\ Compare Art. 124t. 72. Casb II. The boundary of the fluid consists of a rigid cylindrical surface rotating with angular velocity a> about an axis parallel to its length. Taking the origin in the axis of rotation, and the axes of a;, y in a per- pendicular plane, then, with the same notation as before, d^jds will be equal to the normal component of the velocity of the boundary, or dJt dr if r denote the radius vector from the origin. Integrating we have, at all points of the boundary, i/f = ^r* -f const (1) If we assume any possible form of ^, this will give us the equation of a series of curves, each of which would, by rotating round the origin, produce the system of stream-lines determined by tf/. As examples we may take the following : r. If we assume ^= Ar^ coa2e = A {x^ - y% (2) the equation (1) becomes (icu - ^) x2 -f (icu -f A) y^ = C, which, for any given value of 4, represents a system of similar conies. That this system may include the elUpse * Prof. Hole Shaw has made a number of beautiful experimental delineations of the forms of the stream -lines in cases of steady irrotational motion in two dimensions, including those figured on pp. 76, 81; see Trans, Inat, Nav. Arch. t. xl. (1898). The theory of his method will find a place in Chapter xi. t When the general direction of the stream makeB an angle a with the lamina the couple is i^P9o*c* sin 2a. Cisotti, Ann. di mat. (3), t. xix. p. 83 (1912). 71-72] Botating Bomidary 83 we must have ( Ja> — A)a*= (J<«i + A) 6*, Hence the formula ^ = Jcu . - ^ ,^ (x* — y*) . . . 5r (3) gives the motion of a liquid contained within a hollow cylinder whose section is an ellipse with semi-axes a, 6, produced by the rotation of the cylinder about its longitudinal axis with angular velocity co. The arrangement of the stream-lines ^ = const, is shewn in the figure on p. 85. The corresponding formula for <f> is a* - 6« ^^-^-^njTp-^ W The kinetic energy of the fluid, per unit length of the cylinder, is given by ^^ - "//Kir - 01 "»=» ^^■"— '«'•••• ■<») This is less than if the fluid were to rotate with the boundary, as one rigid mass, in the ratio of W + bV to unity. We have here an illustration of Lord Kelvin's minimum theorem, proved in Art. 45.- 2^. Let us assume ^ = 4r» cos 3d = -4 (x» - 3ay*). The equation (1) of the boundary then becomes ^ (x» - 3xy«) - icu (a;« + y2) = C (6) We may choose the constants so that the straight hne x=^ a shall form part of the boundary. The conditions for this are Aa^ - iwa^ = C, ^Aa 4- Jo) = 0. Substituting the values of Ay C hence derived in (6), we have a^ - a» - ^xy^ -h 3a (x^ - a« + y^) = 0. Dividing out by x — a, we get x^+iax^- 4a« = 3y«, or a; 4- 2a = ± \/3 • V- The rest of the boundary consists therefore of two straight lines passing through the point (— 2a, 0), and inclined at angles of 30° to the axis of x. We have thus obtained the formulae for the motion of the fluid contained within a vessel in the form of an equilateral prism, when the latter is rotating 6—2 84 Motion of a Liquid in Two Dimensions [chap, iv with angular velocity w about an axis parallel to its length and passing through the centre of its section; viz. we have V'^-i^^cosSi?, <^ = ^~r»sin3d, (7) where 2 \/3a is the length of a side of the prism*. 3°. In the case of a liquid contained in a rotating cylinder whose section is a circular sector of radius a and angle 2a, the axis of rotation passing through the centre, we may assume , „ cos 2d ^^ /|.\(2n+l)ir/2« -g the middle radius being taken as initial line. For this makes ^ = |a>r^ for d = ± a, and the constants -^jn+i can be determined by Fourier's method so as to make ^ = ^wd^ for r = a. We find ^«n+i = (-)»+^ oM* l(2n+l)^-4a - (2n+l)fl- ^ (2n+l)^ + 4a} ' * " <^^ The conjugate expression for ^ is The kinetic energy is given by 2T=~p|<^^<fe=~ 2pcjj\rdr, (11) where <f>a denotes the value of <f>ioT = a, the value of d<f>/dn being zero over the circular part of the boundary f. The case of the semicircle a = Jtt will be of use to us later. We then have a>a2 f 1 2 1 ^2n+l — {"^r^ - 1o^ 1 " O*, 1 1 "^ TT I2n - 1 2n 4- 1 2n + 3 and therefore (12) J ^-^^'" ^ T" ^ 2;r+3 12;^^^: ■" 2im "^ 2;rT3! ^ " "^ l^ " s" J • Hence J 27 = J7r/)a>«a* (^^ " i) = 'SlOGa^ x J7r/)a>^2 (13) This is less than if the fluid were solidified, in the ratio of *6212 to 1. See Art. 45. * The problem of fluid motion in a rotating cylindrical case is to a certain extent mathe- matically identical with that of the torsion of a uniform rod or bar. The above examples are mere adaptations of two of de Saint- Venant*8 solutions of the latter problem. See Thomson and Tait, Art. 704 et seq, t This problem was first solved by Stokes, " On the Critical Values of the Sums of Periodic Series," Camb. Trane. t. viii. (1847) [Papers, t. i. p. 306]. See also Hicks, Jfe**. of Math, t. viii. p. 42 (1878); Greenhill, (bid. t. viii. p. 89, and t. x. p. 83. X Greenhill, l.c. 72] Rotating Cylinder 85 4°. With the same notation of elliptic co-ordinates as in Art. 71, 3°, let us assume ^ + i^ = (7i6-2(^+*^> (14) Since x* + y* = \c^ (cosh 2^ + cos 217), the equation (1) becomes Ce"*^ cos 2?^ — Jcuc^ (cosh 2^ 4- cos 2iy) *= const. This system of curves includes the ellipse whose parameter is |^o> provided Cc-afo ^ jeuc* = 0, or, using the values of a, 6 already given, C = icu (a + h)\ 80 that ^ = i<o (a + 6)^ e"^ cos 2?^, ' ^ = icu (a + 6)^ c-2f sin 2^7. j At a great distance from the origin the velocity is of the order l/r*. The. above formulae therefore give the motion of an infinite mass of liquid, otherwise at rest, produced by the rotation of an elliptic cylinder about its axis with angular velocity co*. The diagram shews the stream-lines both inside and outside a rigid eUiptical cylindrical case rotating about its axis. (15) The kinetic energy of the external fluid is given by 2T = ^pc* . a>2 (16) It is remarkable that this is the same for all confocal elliptic forms of the section of the cylinder. * Qiutri, Joum. Math. t. xiv. (1875); see also Beltrami, Lc, ante p. 81. 86 Motion of a Liquid in Two Dimefnsions [chap, iv Combining these results with those of Arts. 66, 71 we find that if an elliptic cylinder be moving with velocities J7, V parallel to the principal axes of its cross-section, and rotating with angular velocity a>, and if (further) the fluid be circulating irrotationally round it, the cyclic constant being ic, then the stream-function relative to the aforesaid axes is The foihs followed by the particles of fluid, as distinguished from the stream-lines, in several of the preceding cases, have been studied by Prof. W. B. Morton f; they are very remarkable. The particular case of the circular cylinder (Art. 68) was examined by Maxwell]:. Steady Motions vnth a Free Surface. 73. The first solution of a problem of two-dimensional motion in which the fluid is bounded partly by fixed plane walls, and partly by surfaces of constant pressure, was given by Helmholtz§. KirchhoS|| and others have since elaborated a general method of dealing with such questions. If the surfaces of constant pressure be regarded as free, we have a theory of jets, which furnishes some interesting results in illustration of Art. 24. Again, since the space beyond these surfaces may be filled with liquid at rest, without altering the conditions of the problem, we obtain also a number of cases of 'discontinuous motion,' which are mathematically possible with perfect fluids, but whose practical significance is less easily estimated. We shall return to this point at a later stage (Chap, xi) ; in the meantime we shall speak of the surfaces of constant pressure as 'free.' Extraneous forces, such as gravity, being neglected, the velocity must be constant along any such surface, by Art. 21 (2). The method in question is based on the properties of the function C introduced in Art. 65. The moving fluid is supposed bounded by stream- lines iff = const., which consist partly of straight walls, and partly of lines along which the resultant velocity (q) is constant. For convenience, we may in the first instance suppose the units of length and time to be so adjusted that this constant velocity is equal to unity. Then in the plane of the function f the lines for which q = 1 are represented by arcs of a circle of unit radius, having the origin as centre, and the straight walls (since the direction * The case of a oylindrical lamina whose Bection is an arc of a circle, with circulation round it is solved by Kutta, 8itzb, d. k. bayr. Akad, d. Wiss. 1910; some related problems are discussed by Blasius, ZeiUchr.f, Math, u. Phya. t. liz. p. 226 (1911). t Proc. Roy. 8oc. A. t. Ixzxix. p. 106 (1913). t Proc. Lond. Math. 8oc. t. iii. p. 82 (1870) [Papers, t. ii. p. 208]. § Loc. ciL ante p. 21. I) "Zur Theorie freier Flussigkeitsstrahlen," CreUe, t. Ixz. (1869) [Gea. Abh. p. 416]. See also his Mechanik, co. xzi., zxii. 72-73] Free Stream-Lines 87 of the flow along each is constant) by radial lines drawn outwards from the circumference. The points where these lines meet the circle correspond to the points where the bounding stream-lines change their character. Consider, next, the function log ^. In the plane of this function the circular arcs for which j = 1 become transformed into portions of the imaginary axis, and the radial lines into lines parallel to the real axis, since if { = q-^ e^ we have logC = log^ + ie (1) It remains, then, to determine a relation of the form* logC=fM, (2) where w = <f> -\- up, bs usual, such that the rectilinear boundaries in the plane of log ^ shall correspond to straight lines iff = const, in the plane of w. There are further conditions of correspondence between special points, one on the boundary, and one in the interior, of each region, which render the problem determinate. When the correspondence between the planes of ^ and w has been estabUshed, the connection between z and t<7 is to be found, by integration, from the relation £--«• o The arbitrary constant which appears in the result is due to the arbitrary position of the origin in the plane of z. The problem is thus reduced to one of conformal representation between two areas bounded by straight lines f. This is resolved by the method of Schwarz and ChristofEel, already referred to J, in which each area is repre- sented in turn on a half-plane. Let Z {= X -^ iY) and t be two complex variables connected by the relation ^ = 4 (a - t)-^l- {b - t)-^i^ (c - t)-fi^. . . , (4) where a,b, c, ... are real quantities in ascending order of magnitude, whilst a, ]3, y, ... are angles (not necessarily all positive) such that a + P + y^ ... =27r; (5) and consider the line made up of portions of the real axis of t with small semi-circular indentations (on the upper side) about the points a, b, c, If a point describe this line from t = — oo toi= + oo, the modulus only of the expression in (4) will vary so long as a straight portion is being * The use of log i*, in place of tt is dne to Planck, Wied. Ann. t. xxi. (1884). t See Forsyth, Theory of Functions, c. xx, { See the second footnote on p. 78 ante. 88 Motion of a Liquid in Two Dimensions [chap, iv described, whilst the e£Eect of the clockwise description of the semi-circular portions is to introduce factors 6**, e*^, e% ... in succession. Hence, regarding dZ/dt as an operator which converts 8^ into 8Z, we see that the upper half of the plane of t is conformably represented on the area of a closed polygon whose exterior angles are a, j8, y, . . . , by the formula Z = Al(a - t)-^i^ (b-t)-^'^ (c - t)-^l^ ...dt + B, (6) provided the path of integration in the ^plane lies wholly within the region above delimited. When a, by c, . . , , a, j8, y, ... are given, the polygon is completely determinate as to shape ; the complex constants A, B only affect its scale and orientation, and its position, respectively. As already indicated, we are specially concerned with the conformal representation of rectangular areas. If a = j8 = y = 8 = j7r, the formula (6) becomes r dj ^ = ^ J V{(a - t) {b -t)(c- t) (d -t)}'^^ ^^^ It is easily seen that the rectangle is finite in all its dimensions unless two at least of the points a, 6, c, d are at infinity. The excepted case is the one specially important to us; the two finite points may then conveniently be taken to be t = ± 1, so that = A cosh-i t + B (8) In particular, the assumption ^ t = cosh jy (9) where k is real, transforms the space bounded by the positive halves of the lines Y = 0, Y = irk, and the intervening portion of the axis of Y, into the upper half of the plane t, Cf. Art. 66, 1°. Again, if the two finite points coincide, say at the origin of t, we have Z = AJj + B = Alogt-hB (10) This transforms the upper half of the t-plane into a strip bounded by two parallel straight hues. For example, if « = e^/*, (11) where k is real, these may be the lines Y = 0, Y = irk, 74. As a first appUcation of the method in question, we may take the case of a fluid escaping from a large vessel by a straight canal projecting inwards*. This is the two-dimensional form of Borda's mouthpiece, referred to in Art. 24. * This problem was first solved by Helmholtz, Lc. ante p. 21. 73-74] Borda's Mouthpiece 89 The boundaries of corresponding areas in the planes of ^, log ^, and w, respectively, are easily traced, and are shewn in the figures'*'. It remains to connect the areas in the planes of log ^ and w each with the upper half- plane of an intermediate variable L It appears from equations (8) and (10) of the preceding Art. that this is accomplished by the substitutions logC^ A cosh-i t + B, w = Clogt + D (1) We have here made the comers A, A' in the plane of log ^ correspond to t = ±ly and we have also assumed that f = corresponds to i/? = — oo , as is evident on inspection of the figures. To specify more precisely the values of the cyclic functions cosh~^ t and log t we will assume that they both vanish at ^ =: 1, and that their values at other points in the positive half -plane are A -r' / ^ I I ■J' I i t I 2 B' W A J. ^' A' B' determined by considerations of continuity. It follows that when < = — 1 the value of each function will be vrr. At the points A', A in the plane of log ^, we have, on the simplest convention, log ^ = and 2tTr, respectively ; whence, towards determining the constants in (1) we have = 5, 2i7T = vttA -h B, so that log C = 2 cosh-i t (2) Again, in the plane of w we take the line //' as the line ^ = 0; and if the final breadth of the issuing jet be 26, the bounding stream-lines will be iff = ±b. We may further suppose that ^ = is the equipotential curve passing through A and A\ Hence, from (1) = ittC + D, -ib^D, 80 that w = — \ogt — ib (3) * The heavy lines oonespond to rigid boundaries, and the fine continuous lines to free surfaces. Corresponding points io the various figures are indicated by the same letters. 90 Motion of a Liquid in Two Dimensions [chap, iv It is easy to eliminate t between (2) and (3), and thence to find the relation between z and w by integration, but the formulae are perhaps more convenient in their present shape. The course of either free stream-line, say A'ly from its origin at A\ is now easily traced. For points of this Une t is real, and ranges from 1 to 0; we have, moreover, from (2), iff = 2 cosh"* «, or < = cos \6, Hence, also, from (3), ^ = — log cos \9 (4) Since, along this line, we have d<f>ld8 = — j = — 1, we may put^ = — «, where the arc 8 is measured from A\ The intrinsic equation of the curve is therefore 5 = — log sec i& (5) IT From this we deduce in the ordinary way x = -(8in2J0-logsecje), y = -(0-sin0), (6) TT TT if the origin be at A', By giving 6 a series of values ranging from to tt. the curve is easily plotted. Line of Symmetry. Since the asymptotic value of y is 5, it appears that the distance between the fixed walls is 46. The coefficient of contraction is therefore J, in accord- ance with Borda's theory. 75. The solution for the case of fluid issuing from a large vessel by an aperture in a plane wall is analytically very similar. The chief difference is that the values of log ^ at the points A^ A' in the figures must now be taken to be and — iir, respectively, whence, to determine the constants -4, B in (1) we have so that log 5 = cosh-^ t — vrr (7) 74-761 Vena Contracta 91 The relation between w and t is exactly as before, viz. t£7 = — log < — io, IT (8) where 26 is the final breadth of the stream, between the free boundaries. z r I A' M r^ hgt B' -r B' W A -/' For the stream-line -4/, t is real, and ranges from — 1 to 0. Since, also, i6 = cosh"^ ^ — ijT we may put t = cos (d 4- ^), where d varies from to — Jtt . Hence, from (8), with^ = — *, we have, for the intrinsic equation of the stream- line, « = ?^ log (- sec e) (9) TT Line of Symmetry. 92 Motion of a Liquid in Two Dimensions [chap, iv From this we find 46 26 x = — sin* ^0, y = — {log tan (\tt + \0) — sin 0), TT IT (10) if the point A in the first diagram be taken as origin'*'. The curve is shewn (in an altered position) at the foot of the preceding page. The asymptotic value of a?, corresponding to = — Jtt, is 26/7r, the half width of the aperture is therefore {it + 2) 5/Tr, and the coefficient of con- traction is 7r/(7r + 2) = -611. 76. In the next example a stream of infinite breadth is supposed to impinge directly on a fixed plane lamina, and thence to divide into two portions bounded internally by free surfaces. The middle stream-Une, after meeting the lamina at right angles, branches off into two parts, which follow the lamina to the edges, and thence form the free boundaries. Let this be the line = 0, and let us further suppose that at the point of divergence we have ^ = 0. The forms of the boundaries in the various planes are shewn in the figures. The region occupied by the X C! A logl A c A* A A ^ / A' — 1 3_ moving fluid now corresponds to the whole of the plane w, which must however be regarded as bounded internally by the two sides of the line = 0, <^ < 0. With the same conventions as in the beginning of Art. 75, we have log 5 = cosh"^ t — im, (1) or « = -cosh(logO = -j(C+^). (2) * This example was given by Kirchhoff (Z.c.)» and discussed more fully by Rayloigh, "Notes on Hydrodynamics,*' PhiL Mag. Dec. 1876 [Papers, t. i. p. 297]. 75-76] Impact of a Stream on a Lamina 93 The correspondence between the planes of w and t is best established by considering first the boundary in the plane of w~^. The method of Schwarz and ChristofEel is then at once applicable. Putting a = — tt, j8 = y= ... =0, in Art. 73 (4), we have IT = At, w-i = J4^* + B. (3) At I we have < = 0, vr^ — 0, so that S = 0, or (say) C «;=--,. (4) To connect C (which is easily seen to be real) with the breadth (i) of the lamina, we notice that along CA we have { == g~^, and therefore, from (2) «=-i(^ + ?)> g=-<-\/(^«-l), (5) the sign of the radical being determined so as to make 9 = f or ^ = — oo . Also, dxld<f> = — Ijq. Hence, integrating along CA in the first figure we have I J -00 ^f*-*«/: -1 dt qt* 00 = -4C|' J- < + V(<* - 1)}*, • .(6) whence I (7) Line of Symmetry. Along the free boundary AI, we have log { = i^, and therefore, from (2) and (4), t^-cwd, <I> = -Caw*e (8) 94 Motion of a Liquid in Two Dimensions [chap, iv The intrinsic equation of the curve is therefore 8 = — —7 sec^ 6. TT + 4 where 9 ranges from to — \tt. This leads to 21 a? = — , -7 (sec e + i^r), (9) y = I 7r + 4 {sec tan - log (iw + \e)), (10) the origin being at the centre of the lamina. The curve is shewn on the preceding page. The excess of pressure on the anterior face of the lamina is, by Art. 23 (7), equal to Jp (1 — g*). Hence the resultant force on the lamina is '>/;l<'-'^S*--W!l(r')?--W->"'-"?-'"*- (11) It is evident from Art. 23 (7), and from the obvious geometrical similarity of the motion in all cases, that the resultant pressure (Pq? ^^J) ^U vary as the square of the general velocity of the stream. We thus find, for an arbitrary velocity jo*> — V~4. P?o* • ^ = '440/)go* A (12) P« = 77. If the stream be oblique to the lamina, making an angle a, say, with its plane, the problem is modified in the manner shewn in the figures. /y I ■ / / C^A logl o , ? 7 A' W A' W — 1 > • Kirchhoff, l,c. ante p. 86; Rayleigh, "On the Resistance of Fluids," Phil Mag. Dec. 1876 [Papers, 1. 1. p. 287]. 76-77] Pressure on a Lamina 95 The equations (1) and (2) of the preceding Art. still apply; but at the point / we now have f =«~* , and therefore t = cos a. Hence, in place of (4) *, «^=-7i— — ,. (13) (t - cos of At points on the front face of the lamina, we have, since q-'^=\(\, ^= ±f+V(^«-l), g= ±<"V(«*-1) (14) where the upper or the lower signs are to be taken, according as < ^ 0, i.e, according as the point referred to lies to the left or right of C in the first figure. Hence Between A' and C, t varies from 1 to oo, whilst between A and C the range is from - 00 to - 1. If we put 1 - cos a cos a> cos a — cos a> the corresponding ranges of » will be from ir to a, and from a to 0, respectively ; and we find dt cos a - COB « . , . ,/^« 1 V sin a sin ft) rj r»= r-7 SlU ftlOft), ± ^(i^-l) = . (t - COS ay sm* a ^ x / ^^g ^ _ ^^^ ^ Hence j-= - . i (1 - cos a cos m + sin a sin <») sin a>, (16) and therefore a; = . . {2 cos «> +cos a sin* tt> +sin a sin » cos ta+ihr - «) sin a}, (17) where the origin has been adjusted so that x shall have equal and opposite values when «» =0 and « =7r, respectively ; t.e it has been taken at the centre of the lamina. Hence, in terms of C, the whole breadth is /=*+'^.C. (18) The distance, from the centre, of the point (o =a) at which the stream divides is _ 2cosa(l +sin*a)+(^ir -a) sin a . ~ 4+irCOSa To find the total pressure on the front face, we have 4+9rC06a = r-5~ . Sm" ttCfo). • 3 .™ w.-, (20) sm' o ^ ' Integrated between the h'mits ir and 0, this gives TpC/sin' a. Hence, in terms of I, and of an arbitrary velocity gg of the stream, we find ^ ^^rina_ ^ (21) * The solution up to this point was given by Kirchhoff {Crelle, Lc); the subsequent discus- sion is taken, with merely analytical modifications, from the paper by Rayleigh. 96 Motion of a Liquid in Two Dimensions [chap, rv To find the centre of pressure, we take moments about the centre of the lamina. Thus \p 1(1 -(f')xdx= - .^ . / .r8in'a>(2a> 3 8m» a y » ^ irpO Ccosa sin' a * 8in*a ' ^ ' on substituting the value of x from (17). The first factor represents the total pressure; the abscissa x of the centre of pressure is therefore given by the second, or, in terms of the breadth . cos a , x-\- — .— ./. 4 +7rsma (23) In the following table, derived from Rayleigh*s paper, the column I gives the excess of pressure on the anterior face, in terms of its value when a =00°; whilst columns II and III give respectively the distances of the centre of pressure, and of the point where the stream divides, from the centre of the lamina, expressed aa fractions of the total breadth *. a I II ni 90° 1000 •000 •000 70° •965 •037 •232 60° •854 •075 •402 30° •641 •117 •483 20° •481 •139 •496 10° •273 •163 •500 78. An interesting variation of the problem of Art. 76 has been discussed by Bobylefif. A stream is supposed to impinge symmetrically on a bent lamina whose section consists of two equal straight lines forming an angle. If 2a be the angle, measured on the down-stream side, the boundaries in the plane of ^ can be transformed, so as to have the same shape as in the Art. cited, by the assumption provided A and n be determined so as to make f = 1 when f =e"'****~*^ and f sc"*"" when f =€-«*»+•). This gives On the right-hand half of the lamina, i will be negative as before, and since 9~^ = | CL (24) Henoe di <V(«*-1)' rf^ iV(««-l)* * For the compaiison with oxporimental results see Rayleigh, Z.r. and Nature, t. xlv. (1891) [Papers, t. iii. p. 491]. t Journal of the Ruseian Physico-Chemuxd Socieiy, t. xiii. (1881) [Wiedemann's BeibldUcr, t. vi. p. 163]. The problem appears, however, to have been previously discussed ia a similar manner by )L R^thy, Klausenburger Berichte, 1879. It is generalized by Bryan and Jones, Proc, Boy, Soe. A, t. xci. p. 354 (1915). 77-78] Bobykff's Problem 97 These can be reduced to known forms by the substitution where » ranges from to 1. We thus find (26) (26) We have here used the formulae yo(i+«)* yoi + « / 71— — r=a« =-*+«/ = a«j>, jo(i+«)* ' yoi+» where 1 > ifc > 0. Since, along the stream-line, d^ldtf) = - l/g, we have from (25), if 6 denote the half- breadth of the lamina. . ^f, 2a 4a« P4>^'' , 1 (27) The definite integral which occurs in this expression can be calculated from the formula !fH'^- ii-t)'(8-t) ^**"-w-i*«-<') m / where "^ (m), =d/dfn . log n (m), is the function introduced and tabulated by Gauss*. The normal pressure on either half is, by the method of Art. 76, ^'^ J ^oo \q J at " '^jol+w '^sm Jwir '^ ir sm a The resultant pressure in the direction of the stream is therefore 4a* (29) ir pC, (30) Hence, for any arbitrary velocity Qq of the stream, the rteultant pressure is P^^.pqo^b (31) where L stands for the numerical factor in (27). For a =in, we have L =2 + jfr, leading to the same result as in Art. 76 (12). In the following table, taken (with a slight modification) from Bobyleff's paper, the second column gives the ratio P/Pq of the resultant pressure to that experienced by a plane strip of the same area. This ratio is a maximum when a = 100°, about, the lamina being then concave on the up-stream side. In the third column the ratio of P to the dis- tance (26 sin a) between the edges of the lamina is compared with ipQoK For values of a nearly equal to 180°, this ratio tends to the value unity, as we should expect, since the fluid within the acute angle is then nearly at reflt, and the pressure-excess therefore practically * '* DisqaisitioQos generalos circa seriem infinitam ...,"' Wtrire, Qottingen, 1S70 . . . , t. iii. p. 161. L. H. 7 98 Motion of a Liquid in Two Dimensions [chap, iv equal to ipqo** The last column gives the ratio of the resultant pressure to that experi- enced by a plane strip of breadth 26 sin a, as calculated from (12). a PIP, Plpq^h sin a P/Po8ina 10° •030 •199 •227 20° •140 •359 •409 30° •278 •489 •555 40° •433 •593 •674 45° •612 •637 •724 50° •589 •677 •769 60° •733 •745 •846 70° •854 •800 •909 80° •945 •844 •959 90° 1^000 •879 1000 100° 1016 •907 1031 110° •995 •931 1059 120° •935 •950 1079 130° •840 •964 1096 135° •780 •970 1103 140° •713 •975 1109 150° •559 •984 1119 160° •385 •990 1126 170° 1 •197 •996 1132 Discontinuous Motions, 79. It must suffice to have given a few of the more important examples of steady motion with a free surface, treated by what is perhaps the most systematic method. Considerable additions to the subject have been made by Michell*, Lovef, and other writers J. It remains to say something of the physical considerations which led in the first instance to the investigation of such problems. We have, in the preceding pages, had several instances of the flow of a Uquid round a sharp projecting edge, and it appeared in each case that the velocity there was infinite. This is indeed a necessary consequence of the assumed irrotational character of the motion, whether the fluid be incom- pressible or not, as may be seen by considering the configuration of the « "On the Theory of Free Stream-lines/' Phil, Trans, k, t. clxxxi. (1890). t '*0n the Theory of Discontinuous Fluid Motions in Two Dimensions," Proc, Camb, Phil, Soc, t, viL (1891). X For references see Love, Encycl. d. maih, Wi«s. t. iv. (3), pp. 97 ... . A very complete aoconnt of the more important known solutions, with fresh additions and developments, is given by GreenhiU, Report on the Theory of a Stream-line past a Plane Barrier, published by the Advisory Committee for Aeronautics, 1910. The extension to the case of curved rigid boundaries is discussed in a general manner in various papers by Levi-Civita and OisottL For these, reference may be made to the Rend, d, Circolo Mat, di Palermo, tt, xxiii. xxv. xxvi. xxviii and the Rend, d. r, Accad, d, Lincei, tt. xx. xxi. ; the working out of particular cases naturally presents great difficulties. The theory of mutually impinging jets is treated very fully by Gisotti, " Vene confluenti," Ann, di mat, (3), t. xxiii. p. 285 (1914). 78-79] Discontinuous Motions 99 equipotential surfaces (which meet the boundary at right angles) in the immediate neighbourhood. The occurrence of infinite values of the velocity may be avoided by supposing the edge to be slightly rounded, but even then the velocity near the edge will much exceed that which obtains at a distance great in comparison with the radius of curvature. In order that the motion of a fluid may conform to such conditions, it is necessary that the pressure at a distance should greatly exceed that at the edge. This excess of pressure is demanded by the inertia of the fluid, which cannot be guided round a sharp curve, in opposition to centrifugal force, except by a distribution of pressure increasing with a very rapid gradient outwards. Hence, unless the pressure at a distance be very great, the maintenance of the motion in question would require a negative pressure at the comer, such as fluids under ordinary conditions are unable to sustain. To put the matter in as definite a form as possible, let us imagine the following case. Let us suppose that a straight tube, whose length is large compared with the diameter, is fixed in the middle of a large closed vessel filled with frictionless liquid, and that this tube contains, at a distance from the ends, a sUding plug, or piston, P, which can be moved in any required manner by extraneous forces appUed to it. The thickness of the walls of the tube is supposed to be small in comparison with the diameter ; and the edges, at the two ends, to be rounded off, so that there are no sharp angles. Let us further suppose that at some point of the walls of the vessel there is a lateral tube, with a piston Q, by means of which the pressure in the interior can be adjusted at will. 7—2 100 Motion of a Liquid in Two Dimensions [chap, iv Everything being at rest to begin with, let a slowly increasing velocity be communicated to the plug P, so that (for simplicity) the motion at any instant may be regarded as approximately steady. At first, provided a sufficient force be applied to Q, a continuous motion of the kind indicated in the diagram on p. 72 will be produced in the fluid, there being in fact only one type of motion consistent with the conditions of the question. As the acceleration of the piston P proceeds, the pressure on Q may become enormous, even with very moderate velocities of P, and if Q be allowed to yield, an annidar cavity will be formed at each end of the tube. It is not easy to make out the further course of the motion in such a case from a theoretical standpoint, even in the case of a 'perfect' fluid. In actual liquids the problem is modified by viscosity, which prevents any slipping of the fluid immediately in contact with the tube, and must further exercise a considerable influence on such rapid differential motions of the fluid as are here in question. As a matter of observation, the motions of fluids are often found to differ widely, under the circumstances supposed in each case, from the types represented in such diagrams as those of pp. 71, 72, 80, 81. In such a case as we have just described, the fluid issuing from the mouth of the tube does not immediately spread out in all directions, but forms, at all events for some distance, a more or less compact stream, bounded on all sides by fluid nearly at rest. A familiar instance is the smoke-laden stream of gas issuing from a chinmey. In all such cases, however, the motion in the immediate neighbour- hood of the boundary of the stream is found to be wildly irregular*. It was the endeavour to construct types of steady motion of a frictionless liquid, in two dimensions, which should resemble more closely what is observed in such cases as we have referred to, that led Helmholtzf and Kirchhoff t to investigate the theory of free stream-lines. It is obvious that we may imagine the space beyond a free boundary to be occupied, if we choose, by liquid of the same density at rest, since the condition of constant pressure along the stream-line is not thereby affected. In this way the problems of Arts. 76, 77, for example, give us a theory of the pressure exerted on a fixed lamina by a stream flowing past it, or (what comes to the same thing) the resistance experienced by a lamina when made to move with constant velocity through a liquid which would otherwise be at rest. The question as to the practical validity of this theory will be referred to later in connection with some related problems (Chapter xi.). * Recent experiments would indicate that jets may be formed htfore the limiting velocity of Helmholtz ia reached, and that viscosity plays an essential part in the process. Smoluchowski. **Sur la formation des veines d'efflux dans les liquides/' BuU. de VAcad, de Cracovie, 1904. t U. c. ante pp. 21, 86. 79-80] Flow in a Curved Stratum 101 Fhw in a Curved Stratum, 80. The theory developed in Arts. 59, 60 may be readily extended to the two-dimensional motion of a curved stratum of liquid, whose thickness is small compared with the radii of curvature. This question has been discussed, from the point of view of electric conduction, by Boltzmann*, Kirchhofff* ToplerJ, and others. As in Art. 59, we take a fixed point A, and a variable point P, on the surface defining the form of the stratum, and denote by \fs the flux across any curve AP drawn on this surface. Then ^ is a function of the position of P, and by displacing P in any direction through a small distance Ss, we find that the flux across the element 8^ is given by d^jds . hs. The velocity perpen- dicular to this element will be hfsjhhsy where h is the thickness of the stratum, not assumed as yet to be imiform. If, further, the motion be irrotational, we shall have in addition a velocity- potential <f>, and the equipotential curves ^ = const, will cut the stream-lines = const, at right angles. In the case of uniform thickness, to which we now proceed, it is convenient to write ^ for ^/A, so that the velocity perpendicular to an element hs is now given indifferently by dilsjds and d<f>idn, Sn being an element drawn at right angles to hs in the proper direction. The further relations are then exactly as in the plane problem ; in particular the curves <f> = const., ^ = const., drawn for a series of values in arithmetic progression, the common difference being infinitely small and the same in each case, will divide the surface into elementary squares. For, by the orthogonal property, the elementary spaces in question are rectangles, and if Ssi, Ss^ be elements of a stream-fine and an equipotential fine, respectively, forming the sides of one of these rectangles, we have 3^/3*2 = 3^/9*i> whence 8«i = 8«2, since by construction 80 = &^. Any problem of irrotational motion in a curved stratum (of uniform thickness) is therefore reduced by orthomorphic projection to the corre- sponding problem in piano. Thus for a spherical surface we may use, among an infinity of other methods, that of stereographic projection. As a simple example of this, we may take the case of a stratum of uniform depth covering the surface of a sphere with the exception of two circular islands (which may be of any size and in any relative position). It is evident that the only (two- dimensional) irrotational motion which can take place in the doubly-connected space occupied by the fluid is one in which the fluid circulates in opposite • Wiener Sitzunggberichte, t. lii. p. 214 (1866) [WissenschafUiche Abhandlungen, Leipzig, 1909, t. i p. 1]. t BerL Monai^er. July 19, 1876 [Oes, Abh. p. 66]. t Pogg^ Ann. t. clx. p. 376 (1877). 102 Motion of a Liquid in Two Dimensions [chap, iv directions round the two islands, the cyclic constant being the same in each case. Since circles project into circles, the plane problem is that solved in Art. 64, 2°, viz. the stream-lines are a system of coaxal circles with real 'limiting points' (^, jB, say), and the equipotential lines are the orthogonal system passing through A^ B. Eetuming to the sphere, it follows from well- known theorems of stereographic projection that the stream-lines (including the contours of the two islands) are the circles in which the surface is cut by a system of planes passing through a fixed line, viz. the intersection of the tangent planes at the points corresponding to A and B, whilst the equipotential lines are the circles in which the sphere is cut by planes passing through these points*. In any case of transformation by orthomorphic projection, whether the motion be irrotational or not, the velocity (d^/dn) is transformed in the inverse ratio of a linear element, and therefore the kinetic energies of the portions of the fluid occupying corresponding areas are equal (provided, of course, the density and the thickness be the same). In the same way the circulation {fdift/dn . ds) in any circuit is unaltered by projection. * ThiB example is given by Kirohhoff, in the eleotrical interpretation, the problem considered being the distribution of current in a uniform spherical conducting sheet, the electrodes being situate at any two points A,Boi the surface. CHAPTER V IRROTATIONAL MOTION OF A LIQUID: PROBLEMS IN THREE DIMENSIONS 81. Of the methods available for obtaining solutions of the equation VV = (1) in three dimensions, the most important is that of Spherical Harmonics. This is especially suitable when the boundary conditions have relation to spherical or nearly spherical surfaces. ^ For a full account of this method we must refer to the special treatises*, but as the subject is very extensive, and has been treated from different points of view, it may be worth while to give a shght sketch, without formal proofs, or with mere indications of proofs, of such parts of it as are most important for our present purpose. It is easily seen that since the operator V^ is homogeneous with respect to X, y, z, the part of <f> which is of any specified algebraic degree must satisfy (1) separately. Any such homogeneous solution of (1) is called a * spherical solid harmonic' of the algebraic degree in question. If <^„ be a spherical solid harmonic of degree n, then if we write <l>n = r-S,. (2) S„ will be a function of the direction (only) in which the point (x, y, z) lies with respect to the origin ; in other words, a function of the position of the point in which the radius vector meets a imit sphere described with the origin as centre. It is therefore called a 'spherical surface harmonic' of order nf. ♦ Todhunter, Functions oj Laplace, Lamij and Bessel, Cambridge, 1876. Ferrers, Spherical Harmonics, Cambridge, 1877. Heine, Handbuch der Kugdfiinciionen, 2nd ed., Berlin, 1878. Thomson and Tait, Nahtral Philosophy, 2nd ed., Cambridge, 1879, t. i. pp. 171-218. Byerly, Fourier*9 Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, Boston, U.S.A. 1893. Whittaker, Modem Analysis, Cambridge, 1902. For the history of the subject see Todhonter, History of the Theories of Attraction, dec, Cambridge, 1873, t. 11. Also Wangerin, *'Theorie d. Kugelfmiktionen, n.s.w./* Encyd. d. math. Wiss. U ii. (1) (1904). t The symmetrical treatment of spherical solid harmonics in terms of Cartesian co-ordinates was introduced by Clebsoh, in a much neglected paper, Crelle, t. Izi. p. 195 (1863). It was adopted independently by Thomson and Tait as the basis of their exposition. 104 Irrotational Motion of a Liquid [chap, v To any solid harmonic <^„ of degree n corresponds another of degree — w — 1, obtained by division by r*"+^ ; i.e. <f> = T'-^^-^<f>n is ako a solution of (1). Thus, corresponding to any spherical surface-harmonic S„, we have the two spherical solid harmonics r^S„ and r^"^iS„ . 82. The most important case is when n is integral, and when the surface- harmonic 8^ is further restricted to be finite over the unit sphere. In the form in which the theory (for this case) is presented by Thomson and Tait, and by Maxwell*, the primary solution of (1) is <f>-i-Alr (3) This represents as we have seen (Art. 56) the velocity-potential due to a point-source at the origin. Since (1) is still satisfied when <f> is differ- entiated with respect to a?, y, or 2, we derive a solution ^-« = ^(^ai + ^a^-^^3i)r (*) dy This is the velocity-potential of a double-source at the origin, having its axis in the direction (^ m, n); see Art. 56 (3). The process can be continued, and the general type of spherical solid harmonic obtainable in this way is 3" 1 hi ^8 9 ^8 being arbitrary direction-cosines. This may be regarded as the velocity-potential of a certain configuration of simple sources about the origin, the dimensions of this system being small compared with r. To construct this system we premise that from any given system of sources we may derive a system of higher order by first displacing it through a space ^hg in the direction (Z,, m^, w^), and then superposing the reversed system, supposed displaced from its original position through a space ih, in the opposite direction. Thus, beginning with the case of a simple source at the origin, a first appUcation of the above process gives us two sources 0+, 0_ equidistant from the origin, in opposite directions. The same process applied to the system 0+, 0_ gives us four sources 0++, O.^., 0+_, at the comers of a parallelogram. The next step gives us eight sources at the corners of a parallelepiped, and so on. The velocity-potential, at a great distance, due to an arrangement of 2^ sources obtained in this way, will be given by (5), if hrA = mfhyh^ . . , h„, m' being the strength of the original source at 0. The formula becomes exact, for all distances r, when hi, hz, . , . h„ are diminished, and m' increased, indefinitely, but so that A is finite. * ElectrieUy and Magnetism, c. iz. 81-83] Spherical Harmonics 105 The surface-harmonic corresponding to (5) is given by 3» 1 Sn = i4f"+^ - - -, (6) and the complementary solid harmonic by By the method of * inversion*,' applied to the above configuration of sources, it may be shewn that the solid harmonic (7) of positive degree n may be regarded as the velocity-potential due to a certain arrangement of 2" simple sources at infinity. The lines drawn from the origin in the various directions (Z,, m,, w,) are called the * axes' of the solid harmonic (5) or (7), and the points in which these lines meet the unit sphere are called the ^poles' of the surface-harmonic Sn • The formula (5) involves 2n + 1 arbitrary constants, viz. the angular co-ordinates (two for each) of the n poles, and the factor A, It can be shewn that this expression is equivalent to the most general form of spherical surface-harmonic which is of integral order n and finite over the unit sphere f. 83. In the original investigation of Laplace ;{:, the equation V^ = is first expressed in terms of spherical polar co-ordinates r, ^, co, where a: = r cos ^, y = r sin ^ cos a>, 2 = r sin d sin co. The simplest way of effecting the transformation is to apply the theorem of Art. 36 (2) to the surface of a volume-element rW . r sin ^8a> . 8r. Thus the difference of fiux across the two faces perpendicular to r is ^ ( ^ . fW . r sin Qhii\ 8r. Similarly for the two faces perpendicular to the meridian (a> = const.) we find i(M.,8in^8a,.8r)sd, and for the two faces perpendicular to a parallel of latitude (5 = const.) 5- ( — ' ^1^ . ^85 . 8r ) 8a>. Hence, by addition, This might of course have been derived from Art. 81 (1) by the usual method of change of independent variables. * Explained by Thomson and Tait, Natural PhikMophy, Art. 515. t Sylveeter, PhU, Mag, (5), t. ii. p. 291 (1876) [Mathematical Papers, Cambridge, 1904. .., t. iii. p. 37]. t "Th^rie de Tattraction des sph^roides et de la figure des plan^tes," MAn, de VAtad, toy. des Sciences, 11^ [Oeuvres Computes, Paris, 1878. . ., t. x. p. 341]; Micaniqve CAeOt, Livre 2«»*, c. it 106 Irrotational Motion of a Liquid [chap, v If we now assume that <l> is homogeneous, of degree n, and put ^'^'^'^ s-Ore(«^'^^t) + sT^^^" + «(" + l)^- = «' ••••(2) which is the general differential equation of spherical surface-harmonics. Since the product n (w + 1) is unchanged in value when we write — n — 1 for n, it appears that will also be a solution of (1), as already stated (Art. 81). 84. In the case of symmetry about the axis of a:, the term d^SJdw^ disappears, and putting cos = fi we get ^.{(i-'^*)'^}-'"^"'''^'^"^'' ^'^ the difierential equation of spherical 'zonal' harmonics*. This equation, containing only terms of two different dimensions in /x, is adapted for in- tegration by series. We thus obtain S -aU n (n + 1 ) (n - 2) w (n + 1) (w + 3) ) " " 1 1.2 ^ + 1.2.3.4 ^ ~ • • •) + bL (n-l)(n + 2) (n-8)(n-l)(» + 2)(« + 4) 1 + ^ r ^ 1.2 :~z~ f" + 1.2.3.4.5 1^ "••■[• (2) The series which here present themselves are of the kind called 'hyper- geometric ' ; viz. if we write, after Gauss f, a.jS , a.a+l.j8.j8+l l.y 1.2.y.y-|-l X* ^ a.a+l.a + 2.j8.j8+l./3+2 ^ , ^ 1.2.3.y.y-|-l.y+2 "^ + ' ■ ■ ' (3) we have S„ = AF (- in, i + in,i, fx^) + BfiF (i - in, 1 + fw, f , ^«). . . (4) The series (3) is of course essentially coDvergent when x lies between and 1 ; but when ar = 1 it is convergent if, and only if, y-a-/3>0. In this case we have where n (m) is in Gauss's notation the equivalent of Euler's r (w + 1). * So called by Thomson and Tait, because the nodal lines {8^=(i) divide the unit sphere into parallel belts, t I'C, ante p. 97. 83-85] Zmial Harmonics 107 The degree of divergenoe of the series (3) when y-a-^<0, as X approaches the value 1, is given by the theorem * J^(a,/3,y,x)=(l-a:)y— ^Jf'(y-a,y-Ay,a;). (6) Since the latter series will now be convergent when a; = 1, we see that F (a, ft y, x) becomes divergent as (1 -a;)^"*"""; more precisely, for values of x infinitely nearly equal to unity, we have -P ta, AJ. y, ar) - n (a - 1) . n ()3 - 1) ^* ' ^^^ ultimately. For the critical case where y - a - /3 = 0, we may have recourse to the formula ^^(a,i8,y,x) = ^J^(a + l,/3 + l,y + l,a-), (8) which, with (6), gives in the case supposed ~J'(a,/3,y,ar)=^(l-a;)-i.J^(y-a,y-fty + l,a:) = ^ (1 -ar)-i . J^ (a, 3, a +0 + 1, ar) (9) The last factor is now convergent when a; = 1, so that F {a, fi, y, x) is ultimately divergent as log ( 1 - a;). More precisely we have, for values of x near this limit, i-(«,ft«+A.)=--^iL<^j-;)^iog^ (10) 85. Of the two series which occur in the general expression Art. 84 (2) of a zonal harmonic, the former terminates when n is an even, and the latter when n is an odd integer. For other values of n both series are essentially convergent for values of fi between ± 1, but since in each case we have y — a — j3 = 0, they diverge at the limits /i = ± 1, becoming infinite as log (1 - ,*«). It follows that the terminating series corresponding to integral values of n are the only zonal surface-harmonics which are finite over the unit sphere. If we reverse the series we find that both these cases (n even, and n odd) are included in the formula f p u) = l-3.5...(2n-l) f _ n(n-l) ^nW- i.2.3...n r 2{2n-l)'* I « (n-l)(n-2)(n -3) ) ^ 2.4.(2n-l)(2n-3) ** ...|, ...^i; * Fonyth, Differtniial Equation*, Srd ed., London, 1903, o. vi. t For n even this corresponds to ^ = ( - )*** - * -^ ~- ' ~ » jB =0 ; whilst for n odd we have A =0. B=( -)**"-^> _? — ^' ^:: . See Heine, t. i. pp. 12, 147. 2. 4 ... (»-l> '^'^ 108 Irrotational Motion of a Liquid [chap, v where the constant factor has been adjusted so as to make P^{ji)^\ for /I = 1 *. The formula may also be written The series (1) may otherwise be obtained by development of Art. 82 (6), which in the case of the zonal harmonic assumes the form ^- = ^^"^^a^«r- <^) As particular cases of (2) we have Po(f^) = l, Pi{fi) = f^^ P, (,i) = i {3/^« - 1), Pa (/i) = i (5^» - 3,x). Expansions of P^ in terms of other functions of as independent variables, in place of fi, have been obtained by various writers. For example, we have PAcos0) = l-^'^i'^sin^e + ^^'-^^ (4) This may be deduced from {2)t, or it may be obtained independently by putting /Lt = 1 — 22 in Art. 84 (1), and integrating by a series. The function P^ (fi) was first introduced into analysis by LegendreJ as the coefficient of h^ in the expansion of The connection of this with our present point of view is that if <^ be the velocity-potential of a unit source on the axis of a; at a distance c from the origin, we have, on Legendre's definition, for values of r less than c, 1 r r^ = - + Pi4 + P,l^ + (5) Each term in this expansion must separately satisfy V^ =0, and therefore the coefficient Pn must be a solution of Art. 84 (1). Since P», as thus defined, is obviously finite for all values of /i, and becomes equal to imity for /a = 1, it must be identical with (1). * The functions P^. P,, ... Pj were tabulated by Qlaisher, for values of /i at intervals of *01, BrU, Am, Beport, 1879, and are reprinted by Dale, Five-Figure Tables,.., London, 1903. A table of the same functions for every degree of the quadrant, calculated under the direction of Prof. Perry, was published in the PhiL Mag. for Dec. 1891. Both tables are reproduced in Byerly's treatise, also by Jahnke and Emde, Funktionentafeln, Leipzig, 1909. The values of the first 20 zonal harmonics, at intervals of 5°, have recently been published by Prof. A. Lodge, Phil Trans. A, t. ooiiL (1904). t Murphy, Elementary Principles of the Theories of Electricity, <fcc., Cambridge, 1833, p. 7. [Thomson and Tait, Art. 782.] X "Sur Tattraotion des sph^roides homogdnes," Mim. des Savans jStrangers, t. z. (1786). 86-86] Zonal Harmonies 109 For values of r greater than c, the correeponding expansion is Ice' 4ir<^=:-+Pij5+P,^ + (6) We can hence deduce expressions, which will be useful to us later. Art. 98, for the velocity-potential due to a davble-source of unit strength, situate on the axis of x at a dis- tance c from the origin, and having its axis pointing from the origin. This is evidently equal to 8<^/dc, where <f> has either of the above forms ; so that the required potential is , for r <Cy -4^(^-^2p4+3P.J-...), (7) and for r>c, ^{Pr^^2P,^^...) (8) The remaining solution of Art. 84 (1), in the case of n integral, can be put into the more compact form* <?n(A^) = i^«(A^)log]^-Z„, (9) where Z„ = ^j^ P„_x + g^^^j ^«-8 + (10) This function Q„{n) is sometimes called the zonal harmonic 'of the second kind.' Thus <?o(/*) = ilog}^A Q, (/.) = J (3,i« - 1) log ^ - f /^ Q, 0*) = iM log 5^ - 1, e8(M) = i(V-3/*)logi^-$M' + f- 86. When we abandon the restriction as to symmetry about the axis of X, we may suppose /S„, if a finite and single- valued function of a>, to be expanded in a series of terms varying as cos sw and sin scj respectively. If this expansion is to apply to the whole sphere (i.e. from co = to co = 27r), we may further (by Fourier's theorem) suppose the values of « to be integral. The difEerential equation satisfied by any such term is 5i{<'-'"'fl + {"<"+'>-r^}«"-« <■> If we put iS„ = (l-^«)*%, this takes the form a -H'*)^^-^{'>+l)l^^^ + in- 8)(n + 8 + l)v = 0, ♦ This is equivalent to Art. 84 (4) with, for n even, 4 =0, B=( - )***!— i— V - , ; whilst for n odd we have i4 = ( - )* ^"+^* 2.4. ..(n-1) ^ ^q g^ ^^^^^ ^ ^ pp j^^^ j^^ o , o , , » n 110 Irrotational Motion of a Liquid [chap, t which is suitable for integration by series. We thus obtain 1.2 (n - < - 2) (n - <) (n + * + 1) (n + < + 3) ^_ \ + 1.2.3.4 ^ •••} (n - ^ - 3) (w -g-l)(n + g + 2)(n4-^ + 4) ) + 1.2.3.4.5 '^ •"•]' "^^^ the factor cos «co or sin «a> being for the moment omitted. In the hyper- geometric notation this may be written S„ = (1 - ii})^'{AF (\8 - Jn, i + J« + in, i, /i«) + BfiF (J + i« - Jn, 1 + J5 + in, f , ^«)}. . . .(3) These expressions converge when yi} < 1, but since in each case we have the series become infinite as (1 — /**)*"* at the limits f* = ± 1, unless they terminate*. The former series terminates when n — « is an even, and the latter when it is an odd integer. By reversing the series we can express both these finite solutions by the single formula f n (/^)-2n(n-5)!n!^^ ^^ T 2 . (2n - 1) ^ + (n - g) (n - g - 1) (n - j? - 2) (n - g - 3) „_ n— *— 4 2 . 4 . (2n - 1) (2n - 3) On comparison with Ait. 85 (1) we find that - ...l. ..(4) P„.(M) = (1 -,*«)*• *^^ (6) That this is a solution of (1) may of course be verified independently. In terms of sin i^, we have P • (cos 0) = ^vT — Vi~i sin' ^ U - , ; , , V sm2 i^ " ^ ' 2' (n — «) ! 5 ! ( 1 . (« + 1) * (n - j? - 1) (n - g) (n + g + 1) (n + g + 2) . - ^ _ + ' 1.2.(. + l)(. + 2) 8mM^-...|...(6) This corresponds to Art. 85 (4), from which it can easily be derived. * Rayleigh, Theory of Sound, London, 1877, Art. 338. t There are great yarieties of notation in connection with these * associated functions,* as they have been called. That chosen in the text was proposed by F. Neumann; and is adopted by Whittaker, p. 231. 86-87] Tesseral and Sectorial Harmonics 111 Collecting our results we learn that a surface-harmonic which is finite over the unit sphere is necessarily of integral order, and is further expressible, if n denote the order, in the form tf»n 8^ = A^P^ (ft) + 2 (^,cos 8w 4- 5, sin «cu) P«» (/n), (7) containing 2w + 1 arbitrary constants. The terms of this involving co are called 'tesseral' harmonics, with the exception of the last two, which are given by the formula (1 — /i*)*^ {An cos nco + -B„ sin «a>), and are called 'sectorial' harmonics*; the names being suggested by the forms of the compartments into which the unit sphere is divided by the nodal lines Sn = 0. The formula for the tesseral harmonic of rank s may be obtained otherwise from the general expression (6) of Art. 82 by making n — s out of the n poles of the harmonic coincide at the point = of the sphere, and distributing the remaining s poles evenly round the equatorial circle 5 = Jtt. The remaining solution of (1), in the case of n integral, may be put in the form S„ = (-4, cos «co + B^ sin «co) Q„« (/x), (8) wheret <?.Mf^) == (1 - M*)*'^^^^ (9) This is sometimes called a tesseral harmonic 'of the second kind.' 87. Two surface-harmonics >S, S' are said to be 'conjugate' when JJSS'dnj = 0, (1) where Sto is an element of surface of the unit sphere, and the integration extends over this sphere. It may be shewn that any two surface-harmonics, of different orders, which are finite over the unit sphere, are conjugate, and also that the 2n -h 1 harmonics of any given order n, of the zonal, tesseral, and sectorial types specified in Arts. 85, 86, are all mutually conjugate. It will appear, later, that the conjugate property is of great importance in the physical applications of the subject. Since Sto = sin &S0Sa> = — S/xSeo, we have, as particular cases of this theorem, ' P^(fi)dfjL = 0, (2) /: -1 ri j^^P^(fJL),P,{,JL)dfl=0, (3) * The prefix 'spherical* is implied; it is often omitted for brevity. t A table of'the fanctions Q^ {ft), 0»* (m). for various values of n and s, is given by Brj'an, Proc, Camb, PkU. Soc, t. vi p. 297. 112 Irrotational Motion of a Liquid [chap, v and j[^P„' (jj,) . P„' (ji) d(i = 0. (4) provided m, n are unequal. For m = n, it may be shewn* that / V,W)'«.-2j^ (6) 88. We may also quote the theorem that any arbitrary function f(fAyO)) of the position of a point on the unit sphere can be expanded in a series of surface-harmonics, obtained by giving n all integral values from to 00 , in Art. 86 (7). The formulae (5) and (6) are useful in determining the coefficients in this expansion. Thus, in the case of symmetry about an axis^ the theorem takes the form /(/x) = Co + C^P, (h) + C^P^ (/*)+...+ C^Pn (a^) + (7) If we multiply both sides by Pn (fi) dfi, and integrate between the limits ±1, we find co=ir /o*)««/*. (8) J —X and, generally, Cn = ^^j[jWPnWdfl (9) For the analytical proof of the theorem recourse must be had to the special treatises f; the physical grounds for assuming the possibility of this and other similar expansions will appear, 'incidentally, in connection with various problems. 89. Solutions of the equation V^<f> = may also be obtained by the usual method of treating linear equations with constant coefficients {. Thus, the equation is satisfied by or, more generally, by <f> =/(a» + jSy + yz), (1) provided a* + jS* + y* = (2) * Ferrers, p. 86; Whittaker, pp. 208, 232. t For an account of the more recent investigations of the question, see Wangerin, l.c, X Forsyth, Differential Equations, p. 444 87-89] Integral Formulae 113 For example, we may put a, ^, y = 1, i cos ^, isin ^, (3) or, again, a, j8, y = 1, i cosh w, sinh u (4) It may be shewn* that the most general solution possible can be obtained by superposition of solutions of the type (1). Using (3), and introducing the cylindrical co-ordinates x, ro, a>, where y = m cos o), 2 = ro sin a>, (5) we build up a solution symmetrical about the axis of x if we take 1 r*» <^ = — j f{x + im cos (& — o))} ^. For, since the integration extends over a whole circumference, it is immaterial where the origin of ^ is placed, and the formula may therefore be written f <f> = crl /(aJ4-iwcos&)d^ = - I /(a: + irocosa)(i^. ..(6) ^ This is remarkable as giving a value of ^, symmetrical about the axis of x^ in terms of its values /(x) at points of this axis!]:. It may be shewn, by means of the theorem of Art. 38, that the form of ^ is in such a case completely determined by the values over any finite length of the axis§. As particular cases of (6) we have the functions 1 /•» 1 r» - I (aj + iro cos ^Y cF^, - I (a? 4- its cos &)-'*-^ da, where n will be supposed to be integral. Since these are soUd harmonics finite over the unit sphere, and since, for ro = 0, they reduce to r** and r """^, they must be equivalent to P„ (fi) r»*, and P„ (fi) r-"-^, respectively. We thus obtain the forms ^n (ft) = - r^ + *V(1 - M*) cos a}« (i&, (7) p . X __ 1 r" ^ _ _ _ ^^^ ~ 77 / n lu + CJ(l^ U*i COS ^>+i' ^^ *• "*' 77 ./ (m H- i \/(l - A^*) COS ^}«+i ' due originally to Laplace || and Jacobi^, respectively. • Whittaker, Month, Not. R, AM. Soc. t. Ixii. (1902). t Whittaker, Modern Analysis, c. xiii. t Whittaker, p. 321. § Thomson and Tait Art. 498. I! MSc. Ca. Livre 11—, c. U. t CreUe, t. xxtL (1843) [OesammeUe Werke, Berlin, 1881. . ., t. vi. p. 148], L. H. g 114 Irrotdtional Motion of a Liquid [ceiap. v 90. As a first application of the foregoing theory let us suppose that an arbitrary distribution of impulsive pressure is applied to the surface of a spherical mass of fluid initially at rest. This is equivalent to prescribing an arbitrary value of <f> over the surface ; the value of <f> in the interior is thence determinate, by Art. 40. To find it, we may suppose the given surface- value to be expanded, in accordance with the theorem quoted in Art. 88, in a series of surface-harmonics of integral order, thus ^ = /So + 5i + /Sj+ ... +/S„+ (1) The required value is then <^ = So + ^iS, + ^,S,+ ...-f JiS„+..., (2) for this satisfies V^ = 0, and assumes the prescribed form (1) when r = a^ the radius of the sphere. The corresponding solution for the case of a prescribed value of <^ over the surface of a spherical cavity in an infinite mass of Uquid initially at rest is evidently <A = ^% + ^«i + ^'««+ •.+^>n+ (3) Combining these two results we get the case of an infinite mass of fluid whose continuity is interrupted by an infinitely thin vacuous stratum, of spherical form, within which an arbitrary impulsive pressure is applied. The values (2) and (3) of ^ are of course continuous at the stratum, but the values of the normal velocity are discontinuous, viz. we have, for the internal fluid, dr a ' and for the external fluid ^ = -2(«+l)^. or ^ ' a The motion, whether internal or external, is therefore that due to a distribution of simple sources with surface-density over the sphere ; see Art. 58. 2(2«+l)|' (4) 91. Let us next suppose that, instead of the impulsive pressure, it is the normal velocity which is prescribed over the spherical surface ; thus |^ = S,-hS,+ ...+S„+ ..., (1) 90-92] Applications of Spherical Harmonics 115 the term of zero order being necessarily absent^ since we must have jjfr'^'^' (2) on account of the constancy of volume of the included mass. The value of </> for the internal space is of the form <f> - AjtS^ + ^2^*^2 + . . . + A^r'^S^ + . . . , (3) for this is finite and continuous, and satisfies V^^ = 0, and the constants can be determined so as to make d<f>ldr assume the given surface- value (1) ; viz. we have nAnd^-^ = 1. The required solution is therefore ^ = a2-^iS„ (4) The corresponding solution for the external space is found in like manner to be '^^-'^^dnr-S^- (5) The two solutions, taken together, give the motion produced in an infinite mass of liquid which is divided into two portions by a thin spherical membrane, when a prescribed normal velocity is given to every point of the membrane, subject to the condition (2). The value of <f> changes from dZSn/n to — al,SJ{n + 1), as we cross the membrane, so that the tangential velocity is now discontinuous. The motion, whether inside or outside, is that due to a double-sheet of density _a2^^V^^-; (6) n(n4- 1) see Art. 58. The kinetic energy of the internal fluid is given by the formula (4) of Art. 44, viz. 2T = pjf<f>^dS = pa'lljfs^Hw, (7) the parts of the integral which involve products of surface-harmonics of different orders disappearing in virtue of the conjugate property of Art. 87. For the external fluid we have 2T^-pjj<f>f^dS = pa»l^Jfs„^dm (8) 92. A particular, but very important, case of the problem of the preceding Article is that of t he motio n nf t\ nnti"^ wrhfiffi in **? infinite mass of liquid which is at rest at infinity^ If we take the origin at the centre of the sphere, and the axis of x in the direction of motion, the 8—2 116 Irrotational Motion of a Liquid [chap, v normal velocity at the surface is TJxjr, = U cos d, where U is the velocity of the centre. Hence the conditions to determine <f> are (1°) that we must have V^^ = everywhere, (2°) that the space-derivatives of <f> must vanish at infinity, and (3°) that at the surface of the sphere (r = a) we must have -^=Ucosd (1) I The form of this suggests at once the zonal harmonic of the first order ; we therefore assume The condition (1) gives — 2i4/a* = Z7, so that the required solution is* <f>^\u'^oo^e (2) It appears on comparison with Art. 56 (4) that the motion of the fluid is the same as would be produced by a dauble-source of strength %TVa^, situate at the centre of the sphere. For the forms of the stream-lines see p. 122. To find the energy of the fluid motion we have 2T = - /> /J<^ ^ dS = \paV^ r COS* .27Ta 8m 6 .add = inpa^U^ = M'U^ (3) P^y'3 ii M* = §7r/3a'. It appears, exactly as in Art. 68, that the effect of thie fluid pressure is equivalent simply to an addition to the inertia of the solid, the increment being now half the mass of the fluid displaced f. Thus in the case of rectilinear motion of the sphere, if no external forces act on the fluid, the resultant pressure is equivalent to a force -«'f w in the direction of motion, vanishing when U is constant. Hence if the sphere be set in motion and left to itself, it will continue to move in a straight line with constant velocity. The behaviour of a solid projected in an actual fluid is of course quite different; a continual application of force is necessary to maintain the motion, and if this be not supplied the soUd is gradually brought to rest. * Stokes, "On some oases of Fluid Motion," Camb. Trans, t. Tiii. (1843) [Papers, t. i. p. 17]. Diriohlet, *'Ueber die Bewegung eines festen Korpers in einem inoompressib^ fliissigen Medium,'' Berl Monatsber. 1852 [Werle, Berlin, 1889-97, t. ii. p. 115]. t Stokes, l.e. The result had been obtained otherwise, on the hypothesis of infinitely small motion, by Green, "On the Vibration of Pendulums in Fluid Media," Edin. Trans. 1833^ [Papers, p. 315]. 92-93] Motimi of a Sphere 117 It must be remembered however, in making this comparison, that in a * perfect* fluid there is no dissipation of energy, and that if, further, the fluid be incompressible, the solid cannot lose its kinetic energy by transfer to the fluid, since, as we have seen in Chapter iii., the motion of the fluid is entirely determined by that of the soUd, and therefore ceases with it. 5=^-i«*+^(0 (8) If we wish> to verify the preceding results by direct calculation from the formula P we must remember, as in Art. 68, that the origin is in motion, and that the values of r and for a fixed point of space are therefore increasing at the rates - U cos $, and ((/sin ^)/r, respectively. We thus find, for r^a, ?=Ja^coe^+T^l7«co8 2^-^?7«+JF'(0 (6) The last three terms are the same for surface-elements in the positions 6 and ir -^; so that, when U is constant, the pressures on the various elements of the anterior half of the sphere are balanced by equal pressures on the corresponding elements of the posterior half. But when the motion of the sphere is being accelerated there is an excess of pressure on the anterior, and a defect on the posterior half. The reverse holds when the motion is being retarded. The resultant effect in the direction of motion is -/; JTJ 2ira sin $ .ad$ , p cos $= - §"'pa' -^, as before. 93. The same method can be applied to find the motion produced in a Uquid contained between a solid sphere and a fixed concentric spherical boundary, when the sphere is moving with given velocity U. The centre of the sphere being taken as origin, it is evident, since the space occupied by the fluid is Umited both externally and internally, that soUd harmonics of both positive and negative degrees are admissible ; they are in fact required, in order to satisfy the boundary conditions, which are — d<l>fdr = U cos 6, for r = a, the radius of the sphere, and d<f>/dr = 0, for r = b, the radius of the external boundary, the axis of x being as before in the direction of motion. (B\ Ar + -g j cos 6y (1) and the conditions in question give a' 6' whence ^^^s^' ^^^^^.U (2) 118 Irrotational Motion of a Liquid [chap, v The kinetic energy of the fluid motion is given by the integration extending over the inner spherical surface, since at the outer we have 3<^/3r = 0. We thus find 2r = ^^±^V'C^'. (3) It appears that the effective addition to the inertia of the sphere is now* 6» + 2a« TT 6» - a» />a« (4) As h diminishes from oo to a, this increases continually from ^irpa^ to oo , in accordance with Lord Kelvin's minimum theorem (Art. 45). In other words, the introduction of a rigid spherical partition in the problem of Art. 92 acts as a constraint increasing the kinetic energy for any given velocity of the sphere, and so virtually increasing the inertia of the system. 94. In all cases where t he motion of a liquid takes pl ace in a series o f pla nes passing through a common line, and is Jhej same i n e ach such plane , there exists a f^reft^-jt^nnfyhif^ panaToptviiflln some of its properties to the two- dimensional stream-function of the last Chapter. If in any plane through the axis of symmetry we take two points A and P, of which A is arbitrary, but fixed, wtile P \a variable, then considering the annular surface generated by any line AP, it is plain that the flux across this surface is a function of the position of P. Denoting this function by 27t^, and taking the axis of x to coincide with that of symmetry, we may say that is a function of x and tD, where x is the abscissa of P, and m, = (y* + «*)% is its distance from the axis. The curves ^ = const, are evidently stream-lines. If P' be a point infinitely near to P in a meridian plane, it follows from the above definition that the velocity normal to PP' is equal to 27r8^ 27rtD.PP" whence, taking PP' parallel first to m and then to x, where u and v are the components of fluid velocity in the directions of x and w respectively, the convention as to sign being similar to that of Art. 59. These kinematical relations may also be inferred from the form which the equation of continuity takes under the present circumstances. If we express ♦ Stokes, Lc. ante p. 116. 93-94] Stokes' Stream-Function 119 that the total flux into the annular space generated by the reyolution of an elementary rectangle SxSm is zero, we find ^ {u . ^mhw) See + X- (u . ^TTwhx) hm = 0, or 5- (mu) -I- ^r- Iwv) = 0, (2) which shews that wv .dx — mu . dm is an exact differential. Denoting this by d*ff we obtain the relations (1)*. So far the motion has not been assumed to be irrotational ; the condition that it should be so is dv 9^ _ /^ dx 9t5 ~ ' whichleadsto P^ + ft ]l^ = o (3) ox^ ow^ woto The differential equation of <f} is obtained by writing „=_?^ „=_?^ dx' dm "P).-it» S + ^ + sE-o W It appears that the functions ^ and are not now (as they were in Art. 62) interchangeable. They are, indeed, of different dimensions. The kinetic energy of the liquid contained in any region bounded by surfaces of revolution about the axis is given by -X" . 27Tmd8 ZDOS -=27TpJ4>tb/, (5) 8^ denoting an element of the meridian section of the bounding surfaces, and the integration extending round the various parts of this section, in the proper directions. Compare Art. 61 (2). * The stream-function for the case of symmetry about an axis was introduced in this manner by Stokes. "On the Steady Motion of Incompressible Fluids," Camb. Trans, t. vii. (1842) [Papers, t. i p. 1]. Its analytical theory has been treated very fully by Sampson, "On Stokes' Current- Function," PhU. Trans, A, t. clxxxii. (1891). 120 Irrotational Motion of a Liquid [chap, v 95. The velocity-potential due to a point-source at the origin is of the form <!> = ] (1) The flux thrdugh any closed curve is in this case numerically equal to the solid angle which the curve subtends at the origin. Hence for a circle with Ox as axis, whose radius subtends an angle d at 0, we have, attending to the sign, 2w}t = - 27r (1 - cos fl). Omitting the constant term we have ^ = r = ai (2) The solutions corresponding to any number of simple sources situate at various points of the axis of x may evidently be superposed; thus for the double-source ^ = -air = T« ' (^) , , av nj2 sin^fl ^« l^ave = _ g^^ = _ _ = - -_ (4) And, generally, to the zonal solid harmonic of degree — w — 1, viz. to a** 1 corresponds* == ^ g^^-^ (6) A more general formula, applicable to harmonics of any degree, fractional or not, may be obtained as follows. Using spherical polar co-ordinates r, d, the component velocities along r, and perpendicular to r in the plane of the meridian, are found by making the linear element PP' of Art. 94 coincide successively with rh6 and 8r, respectively, viz. they are f sin fl r3fl' r sin fl 3r ^ ' Hence in the case of irrotational motion we have ~^m-"t |--»l <«) Thus if ^ = f«Sn, (9) where Sn is a zonal harmonic of order n, we have, putting /x = cos 0, * Stefan, "Ueber die Kraftlinien eines urn eine Axe symmetrischen Feldes," Wied, Ann, t. zvii (1882). 95-96] Streani'Lhies of a Sphere 121 The latter equation gives ^ = «il'""(l-'*')f' : <1<^) which must necessarily also satisfy the former; this is readily verified by means of Art. 84 (1). Thus in the case of the zonal harmonic P„, we have as corresponding values ^ = r-P,(/*), ^=-^r»+Ml-M')^". (11) and ^ = r— ip„(/x), ^ = - 1 r- (1 - /*«) '^f" (12) of which the latter must be equivalent to (5) and (6). The same relations hold of course with regard to the zonal harmonic of the second kind, Q„ . 96. We saw in Art. 92 that the motion produced by a solid sphere in an infinite mass of liquid may be regarded as due to a double-source at the centre. Comparing the formulae there given with Art. 95 (4), it appears that the stream-function due to the sphere is = - i Z7 - sin2 e ( 1 ) The forms of the stream-lines corresponding to a number of equidistant values of t/f are shewn on the next page. The stream-Unes relative to the sphere are figured in a diagram near the end of Chapter vil^*23« # Again, the stream-function due to two double-sources having their axes oppositely directed along the axis of x will be of the form ^ = VY -7-8-' (2) where r^, r^ denote the distances of any point from the positions, P^ and Pj, say, of the two sources. At the stream-surface = we have i.e. the surface is a sphere in relation to which P^ and P^ are inverse points. If be the centre of this sphere, and a its radius, we find AjB = OPiVa* = a^lOP^^ (3) This sphere may be taken as a fixed boundary to the fluid on either side, and we thus obtain the motion due to a double-source (or say to an infinitely small sphere moving along Ox) in presence of a fixed spherical boundary. The disturbance of the stream-Unes by the fixed sphere is that due to a double-source of the opposite sign placed at the 'inverse' point, the ratio of 122 Irrotational Motion of a Liquid [chap. V the strengths being given by (3)*. This fictitious double-source may be called the 'image' of the original one. 97. Rankinef employed a method similar to that of Art. 71 to discover forms of solids of revolution which will by motion parallel to their axes generate in a surrounding liquid any given type of irrotational motion symmetrical about an axis. The velocity of the solid being U, and 85 denoting an element of the meridian, the normal velocity at any point of the surface is Udm/ds, and that * This result was given by Stokes, "On the Resistance of a Fluid to two Oaoillating Spheres," Brit Ass. Report, 1847 [Papers, t. i. p. 230]. t **0n the Mathematical Theory of Stream Lines, especially those with Four Foci and upwards," PhiL Trans. 1871, p. 267 (not included in the collection referred to on p. 61 ante). 96^98] Motion of Two Spheres 123 of the fluid in contact is given by — dt/t/wds. Equating these and integrating along the meridian, we have = — iUw^ + const. ^ (1) If in this we substitute any value of t// satisf jnng Art. 94 (3), we obtain the equation of the meridian curves of a series of solids, each of which would by its motion parallel to x give rise to the given system of stream-lines. In this way we may readily verify the solution already obtained for the sphere; thus, assuming ^^Aw^/r^, (2) we find that (1) is satisfied for r = a, provided A=^^^Ua\ : (3) which agrees with Art. 96 (1). 98. The motion of a liquid bounded by two spherical surfaces can be found by successive approximations in certain cases. For two soUd spheres moving in the line of centres the solution is greatly faciUtated by the result given at the end of Art, 96, as to the 'image' of a double-source in a fixed sphere. Let a, 6 be the radii, and c the distance between the centres A, B. Let U be the velocity of A towards B, V that of B towards A, Also, P being any point, let AP=r, BP =/, PAB = 0, PBA =e\ The velocity-potential will be of the form u<t> + u'4>% (1) where the functions and <f>' are to be determined by the conditions that V^<t> =0, ^V =0, (2) P * * ^ ir. I V"ff, throughout the fluid, that their space-derivatives vanish at infinity, and that over the surface of A, whilst ^=-co8tf. ^'=0. (3) ^,=0. g:=— ' (4) over the surface of B. It is evident that is the value of the velocity-potential when A moves with unit velocity towards B, while B \b At rest; and similarly for 0'. 124 Irrotational Motion of a Liquid [ohap. v To find 0, we remark that if B were absent the motion of the fluid would be that due to a certain double-source at A having its axis in the direction AB, The theorem of Art. 96 shews that wc may satisfy the condition of zero normal velocity over the surface of B by introducing a double-source, viz. the 'image* of that at ^ in the sphere B» This image is at Hi, the inverse point of A with respect to the sphere B\ its axis coincides with AB^ and its strength is - yijt^lc^, where /zq is the strength of the original source at A, viz. /^ = 2ira'. The resultant motion due to the two sources at A and H^ will however violate the condi- tion to be satisfied at the surface of the sphere Ay and in order to neutralize the normal velocity at this surface, due to H^, we must superpose a double-source at H^, the image of H^ in the sphere A, This will introduce a normal velocity at the surface of B, which may again be neutralized by adding the image of H2 in B, and so on. If /ix» f4» Ms* • • • be the strengths of the successive images, and/^, f2*h» • • • t^eir distances from Ay we have /x=c-- . ^' "A • Ml" fi" f-"* ^*-/,' M, /," M6~ /."'/ cuid so on, the laws of formation being obvious. The images continually diminish in intensity, and this very rapidly if the radius of either sphere is small compared with the shortest distance between the two surfaces. The formula for the kinetic energy is 2T=-p I l{U<t> + U'<t>') (^^+U' ^^ d8=LU^ +2MUU' +NU'^ (6) provided where the suffixes indicate over which sphere the integration is to be effected. The equality of the two forms of M follows from Green's Theorem (Art. 44). The value of near the surface of A can be written down at once from the results (7) and (8) of Art. 85, viz. we have 4^0=(/io+/*a+/i4 + ...)^^-2fj| + ^+ •..Vcos^+&o., (8) the remaining terms, involving zonal harmonics of higher orders, being omitted, as they will disappear in the subsequent surface-integration, in virtue of the conjugate property of Art. 87. Hence, putting 8</>/8n = - cos B, we find with the help of (5) ^ = ip(;*o+3/ia+3Am-...) = }7r^»(^l-f3^g+3 ^^^3^^_^^^^3^^3 -h...j. ...(9) It appears that the inertia of the sphere Jl is in all cases increased by the presence of a fixed sphere B. Compare Art. 93. The value of N may be written down from symmetry, viz. it is .V =S.p6» (1 +3^1'^ +3 ^^^,, ^f_^J^,^^ -^ + . . . ) (10) 98-99] Motion of Two Spheres 125 where //— ? . f'-c "' ye — f f* ^ (11) and so on. To calculate if we require the value of if/ near the surface of the sphere A ; this is due to double-sources ij^\ /aj', /x,', /ig', ... at distances c, c -/i', c -/g', c -/g', . . . from A^ where /iq' = - 2ir6^, and Ms «» (c -/,')" (12) Hence and so on. This gives, for points near the surface of A, 4ir*' =(/h' +/^' +M5' + . . "^7/ - 2 (^ + ^^^^y^ _« o86»/ a36» a«6« 1 When the ratios a/c and hjc are both small we have L = t7rpa3(l+3^), M^2np^, i^^j^pfts ^1 +3^^^ (15) approximately ♦. If i^ the preceding results we put h=a, U' = U, the plane bisecting AB at right angles will be a plane of symmetry, and may therefore be taken as a fixed boundary to the fluid on either side. Hence, putting c =2h, we find, for the kinetic energy of the liquid when a sphere is in motion perpendicular to a rigid plane boundary, at a distance h from it. 2r=|«-pa»(l+ip + ...)c7* (16) a result due to Stokes. / 99. When the spheres are moving at right angles to the line of centres the problem is more diffictdt; we shall therefore content ourselves with the first steps in the approximation, referring, for a more complete treatment, to the papers cited below. * To this degree of approximation the results may be more easily obtained without the use of * images,* the procedure being similar to that of the next Art. 126 In^otational Motion of a Liquid [chap, v Let the spheres be moving with velocities F, F' in i)arallel directions at right angles to Ay Bj and let r, B, <a and r', 6\ <»' be two systems of spherical polar co-ordinates having their origins at A and B respectively, and their polar axes in the directions of the velocities Vi F^ The velocity-potential will be of the form F« + V'ft>\ with the surface-conditions 5^=-cos^, "i"=^» forr=o, (1) and 5^=0, ^=-cos^', forr'=6 •. (2) or or ^ ' If the sphere B were absent the velocity-potential due to unit velocity of A would be J -^ cos ^. Since r cos 6=r^ cos 0", the value of this in the neighbourhood of B will be J ^ r' cos ^, approximately. The normal velocity at the surface of B, due to this, will be canceUed by the addition of the term a'ft* cos B' which, in the neighbourhood of A becomes equal to I —5- r cos 6, nearly. To rectify the normal velocity at the surface of ^, we add the term Stopping at this points and collecting our results, we have, over the surface of A, 0-i«(l+i^)coB^, (3) a' and at the surface of B, ^ =1^ * is ^^ ^ (^) Hence if we denote by P, Q, R the coefficients in the ezpreraion for the kinetic energy, viz. 2T=PV* +2QVV' +Rr'* (5) we have F= - p JJ^grfS^ =f^pa»(l + J^/ (6) 99-100] Cylindrical Harmonics 127 The oase of a sphere moving parallel to a fixed plane boundary, at a distance A, is obtained by putting 6 =a, V = V\c =2A, and halving the consequent value of T; thus 2r=|,r/K.»(l + A^!)F* (7) This result, which was also given by Stokes, may be compared with that of Art. 98 (16)*. Cylindrical Harmonics. 100. In terms of the cylindrical co-ordinates x, to, co introduced in Art. 89, the equation V*^ = takes the form a*^ av.i?^ ,l!V = o m dx^ 9nj* w dm ro* 9co* This may be obtained by direct transformation, or more simply by expressing that the total flux across the boundary of an element hx ,hm . xnh<i} is zero, after the manner of Art. 83. In the case of symmetry about the axis of x^ the equation reduces to the form (4) of Art. 94. A particular solution is then if> = 6*** x (^)» provided X" (t") + ^ X' («>) + **X («) = (2) This is the differential equation of 'Bessel's Functions' of zero order. Its complete primitive consists, of course, of the sum of two definite functions of tzj, each multipUed by an arbitrary constant. That solution which is finite f or nj = is easily found in the form of an ascending series ; it is usually denoted by CJ© (^)> where .7o(0 = l-|| + 2i^4,- (3) We have thus obtained solutions of V«^ = of the types t ^ = g*t« J^ (km) (4) It is easily seen from Art. 94 (1) that the corresponding value of the stream-function is = If lue*** Jo' i]cm) (5) * For a fuUer analytical treatment of the problem of the motion of two spheres we refer to the following papers: W. M. Hicks, '*0n the Motion of two Spheres in a Fluid," Phil. Trawt, 1880, p. 455; R. A. Herman, "On the Motion of two Spheres in Fluid,** Quart. Jowm, Math, t. zxii. (1887); Basset, ''On the Motion of Two Spheres in a Liquid, fto." Proc. Lond. Math. 8oc. t. zviii. p. 360 (1887). See also C. Neumann, Hydrodynamische Untersuehungen, I^ipzig, 1883; Basset, Hydrodynamics, Cambridge, 1888, t. i. The mutual influence of 'pulsating' spheres, ue, of spheres which periodically change their volume, has been studied by C. A. Bjerknes, with a view to a mechanical illustration of electric and other forces. A full account of these researches is givrai by his son Prof. V. Bjerknes in his VorUsungen Hber hydrodynamische Femkrafte, Leipzig, 1000-1902. The question is also treated by Hicks, Camb. Proc. t. iii. p. 276 (1870). t. iv. p. 29 (1880), and by Voigt, (?dtt. Nachr. 1891, p. 37. t Except as to notation these solutioni* are to be found tn Poisson, l.c. ante p. 17. 128 Irrotatwiial Motion of a Liquid [chap, v The formula (4) may be recognized as a particular case of Art. 89 (6); viz. it is equivalent to g±*(*+.-irco8^) ^^ (6) smce J^(^) = l rcos(Scos^)ci&=- [^ e'^^^g^, (7) as may be verified by developing the cosine, and integrating term by term. Again, (4) may also be identified as the Hmiting form assumed by a spherical solid zonal harmonic when the order (n) is made infinite, provided that at the same time the distance of the origin from the point considered be made infinitely great, the two infinities being subject to a certain relation*. Thus we may take ^ = ^„P„(/.) = (l + ?)"ff,(«,) (8) where we have temporarily changed the meanings of x and to, viz. r = a-\-x^ to = 2a sin |fl, whilst U /n,\ - 1 _ ^ (^ + ^) ^* . (n ~ 1) n (n + 1) (n + 2) ^ . see Art. 85 (4). If we now put h = n\a^ and suppose a and n to become infinite, whilst h remains finite, the symbols x and to will regain their former meanings, and we reproduce the formula (4) with the upper sign in the exponential. The lower sign is obtained if we start with The same procedure leads to an expression of an arbitrary function of m in terms of the Bessel's Function of zero order f. According to Art. 88, an arbitrary function of latitude on the surface of a sphere can be expanded in spherical zonal harmonics, thus J- (,*) = 2 (« + i) P„ (/Lt) J^^ J (/) P„ 1^) dfl' (10) If we denote by w the length of the chord drawn to the variable point from the pole (0 = 0) of the sphere, we have t!j^2asin^d, toSto = — a^8/x, where a is the radius, so that the formula may be written fivj) = ], 2 (n + i) H„ (ID) rf(w') H, (w') w'dm' (11) * This process was indicated, without the restriction to symmetry, by Thomson and Tait, Art. 783 (1867). f The procedure appears to be due substantially to C. Neumann (1862) ; for the history of the theorem (12) see Heine, t. i p. 442, and Niekien (op. eU, p. 129), p. 360. 100-101] Cylindrical Harmonics 129 If we now put * = -, SA = -, and finally make a infinite, we obtain the important theorem : / (to) = r Jo (km) hdkT f (tn') J© (km') w'dw' (12) JO Jo 101. If in (1) we suppose ^ to be expanded in a series of terms varying as cos^cu or sin^co, each such term will be subject to an equation of the form a^ + aS-^ + SaS-Si^""^ ^^^^ This will be satisfied by ^ = e*** x (^)» provided x"M+^x'M + (a«-^)xM = 0, (14) which is the differential equation of Bessel's Functions of order 8*. The solution which is finite for to = may be written x (*) = C'J, (tro), where J^ (0 = 2^ n"(7) r " 2 (2« + 2) "^ 2 . 4 (2« -I- 2) (2« + 4) ""•••}•• -(1^) The complete solution of (14) involves, in addition, a Bessel's Function 'of the second kind' with whose form we shall be concerned at a later period in our subjectf. We have thus Obtained solutions of the equation V*^ = 0, of the types ^ = 6*»« J. (Jfcm) ^.^1 sa> (16) These may also be obtained as limiting forms of the spherical solid harmonics ^'* » w \ cos) a«+i 7> . / X cos] with the help of the expansion (6) of Art. 86 J . * Forsyth, Art. 100; Whittaker, o. zii t For the further theory of the Bessel's Functions of both kinds recourse may be had to Lommel, Studien ueber die BeawPtehen FunkHonenp Leipzig, 1868; Gray and Mathews, Treatise on Bessel Functions, London, 1896 ; H. Weber, PartieUe Differentidlgleiehungen d. nuUh. Physik, Braunschweig, 1900-01; Nielsen, Handbueh d, Theorie d. Cylinderfunkiionen, Leipzig, 1904; and to the treatises of Heine, Todhunter, Forsyth, Byerly, and Whittaker, already cited. An ample account of the subject, from the physical point of view, will be found in Rayleigh's Theory of Sound, oc. ix., zviii., with many important applications. Numerical tables of the functions J, (^) have been constructed by Bessel and Hansen, and more recently by Meissel {BerL Ahh, 1888). Hansen's tables are reproduced by Lommel, and (partially) by Rayleigh and Byerly; whilst Meissel's tables have been reprinted by Gray and Mathews. Abridgments to five and four figures respectively are included in the collections of Dale and of Jahnke and Emde referred to on p. 108. X The connection between spherical surface-harmonics and Bessel's Functions was noticed by Mehler," Ueber dieVertheilung d. statischen Elektricitat in einem v. zwei Kugelkalotten begrenzten L.H. 9 130 Irrotational Motion of a Liquid [chap, v 102. The formula (12) of Art. 100 enables us to write down expressions, which are sometimes convenient, for the value of <f> on one side of an infinite plane {x = 0) in terms of the values of <l> or 3^/3n at points of this plane, in the case of symmetry about an axis (Ox) normal to the plane*. Thus if ^ = J? (id), for a; = 0, (1) we have, on the side x > 0, ^ = r e-*« Jo (*C7) UhT F (tn') Jo (*^) ^'dm' (2) Again, if • "" af "" -^^^^^ for « = 0, (3) we have <f> = I 6-** Jq {*ro) dk I / (to') Jo {Jew') w'dw' (4) The exponentials have been chosen so as to vanish for cr = oo . Another solution of these problems has already been given in Art. 58, from equations (12) and (11) of which we derive ^ = ^-/K©'^' (^) respectively, where r denotes distance from the element hS of the plane to the point at which the value of ^ is required. We proceed to a few applications of the general formulae (2) and (4). 1°. If, in (4), we assume /(ro) to vanish for all but infinitesimal values of ro, and to become infinite for these in such a way that and /, f(m) 27rujdw = i, we obtain 4^ = I 6"** J© (kw) dk, (7) and therefore, since Jq = — Ji, /•oo 47r0 = - tn I e-*» Ji {kw) dk, (8) Jo by Art. 100 (5). Korper," CreUe, t. Ixviii. (1S68). It was investigated independently by Bayleigb, ''On the Relation between the Functions of Laplace and Bessel," Proc. Lond, Math. Soc, t. ix. p. 61 <1878) [Papers, t. i. p. 338]; see also Theory of Sound, Arts. 336, 338. There are also methods of deducing BesseFs Functions 'of the second kind' as limiting cos I forms of the spherical harmonics Q^ (ft), Q^* (/t) . > «»; for these see Heine, t i. pp. 184, 232. * The method may be extended so as to be free from this restriction. 102] Applications of Cylindrical Harmonics 131 By comparison with the primitive expressions for a point-source at the origin (Art. 95), we infer that f e-^ Jo (M d* = J, r «"*« Ji (M dk = ~^-^. , ... .(9) Jo T Ja r [r -\- X) where r = ^/{x^ 4- to*) ; these are in fact known results*. 2^. Let us next suppose that sources are distributed with uniform density over the plane area contained by the circle tn = a, jr = 0. Using the series for Jo, «^i, or otherwise, we find I Jo {km) mdm = j^Ji (ka) (10) .'0 ^ Hence t (11) where the constant factor has been chosen so as to make the total flux through the circle equal to unity. 3°. Again, if the density of the sources, within the same circle, vary as l/\/(^* ~ ®*), w® hskve to deal with the integral I /"^ (M:^;?(J^) = »/*Vo (te sin^) sin ^ d^ = "i^. . .(12) where the evalustion is effected by substituting the series form of Jq, and treating each term separately. Hence (13) if the constant factor be determined by the same condition as before §. It is a known theorem of Electrostatics that the assumed law of density makes (f> constant over the circular area. It may be shewn independently that Jo (kw) sin ia -T- = Jtt, or sin"^ - / / 00 Ji (km) Bmka-j- = ^^ -, or — /cm m (14) * The former is due to Lipsohitz, Crelle, t. Ivi p. 189 (1869); Bee Gray and Mathews, p. 72. The latter follows by di£ferentiation with respect to w and integration with respect to z, t Cf. H. Weber, CreUe, t. Ixxv. p. 88; Heine, t. u. p. 180. X The formula (12) has been given by various writers; see Rayleigh, Papers, t. iiL p. 98; Hobson, Proc. Land. Math. Soe. t. xxv. p. 71 (1893). § CJf. H. Weber, Crdk, t. Ixxv. (1873); Heine, t. ii p. 192. 9—2 132 Irrotational Motion of a Liquid [chap, v according as m ^ a*. The formulae (13) therefore express the flow of a liquid through a circular aperture in a thin plane rigid wall. Another solution will be obtained in Art. 108. The corresponding problem in two dimensions was solved in Art. 66, 1°. 4°. Let us next suppose that when x = 0, we have <f> = C ^/(a* — to*) for tD < a, and ^ = for ro > a. We find I Jq (km) \/(a* — tD*) tDdm = a^ I Jq {ka sin &) sin S^ cos* S^d& = a'^i (fei), Jo Jo (15) provided 0,(O = i(l-^ + 27A77""-:)="|5C^- "^^^^ Hence, by (2), <^ = - cT e'^J^ ^^^M^CT^) ^* ^^"^^ This gives, for x = 0, ^ j = C I Jq (km) sin ia -jT- 4- CtD I Jq {km) sin kadk, . . (18) after a partial integration. The value of the former integral is given in (14), and that of the latter can be deduced from it by difEerentiation with respect to fD. Hence -^).-*'<'-»"'H-S-vi^:^) "" according as id ^ a. It follows that if C = 2/7r . Z7, the formula (17) will relate to the motion of a thin circular disk with velocity U normal to its plane, in an infinite mass of liquid. The expression for the kinetic energy is 2T = - /> |[^ 1^ (iS = 7r/>C* ["* V(a* - in*) 27Tmdm = }7r*/>a»C*, or 2T = ipa^U* (20) The effective addition to the inertia of the disk is therefore 2/7r (= '6366) times the mass of a spherical portion of the fluid, of the same radius. For another investigation of this question, see Art. 108. * H. Weber, Crelle^ t. Ixxv. ; Part Diff,-Ql. t. i p. 189; Gray and Mathews, p. 126. See also Proc» Lond. Math. Soc. t. xxziv. p. 282. 102-104] EUipsoidal Hamwnics 133 EUipsaidal Harmonics. 103. The method of Spherical Hannonics can also be adapted to the solution of the equation VV = 0, (1) under boundary-conditions having relation to ellipsoids of revolution*. Beginning with the case where the ellipsoids are prolate, we write x = k cos cosh Tj — kfi^y y = w cos co, 2 = to sin co,| where to = Asin ffsinhiy = ifc (1 - /i«)* (C« - 1)*. J The surfaces ^ = const., /x = const, are confocal ellipsoids, and hyperboloids of two sheets, respectively, the common foci being the points {± k, 0, 0). The value of ^ may range from 1 to oo , whilst /x lies between ± 1. The co-ordinates fif Ci <*> form an orthogonal system, and the values of the linear elements Stf^y is^, hs^ described by the point {x, y, z) when /i, {, co separately vary are ^^ = * ir^) ^f"' ^'i' * (frzf ) ^^' 8»- = * (1 - /*•)* (i* - 1)* s*"- • \v) To express (1) in terms of our new variables we equate to zero the total flux across the walls of a volume element 8«^S«;S««, and obtain I (a|s.^.) V +1 (|s..8..) H + ^ (|8.A) 8» - 0, or, on substitution from (3), This may also be written dfi 0^ '^ ^a/ij ^ 1 - /i« aco» 35 r ^ ^ acj ^ 1 - c« aco«' • -^^^ 104. If ^ be a finite function of /a and co, from /a=: — lto/A = + l and from CO = to CO » in, it may be expanded in a series of surface harmonics of integral orders, of the types given by Art. 86 (7), where the coefficients are functions of £ ; and it appears on substitution in (4) that each term of the expansion must satisfy the equation separately. Taking first the case of the zonal harmonic, we write ^-P,(/i).Z, (5) * Heine^ "Ueber einige Aufgaben, welohe auf paitielle DifferentialgleiohimgeEi fahren," OreUe, t. xxyi p. 185 (1843), and Kugdfunitionen, t. iL Art. 38. See also Ferrers, o. rl 134 Irrotational Motion of a Liquid [chap, v and on substitution we find, in virtue of Art. 84 (1), ^j i<^ - ^*4|! + " ^" + ^) ^ "^ °' (^) which is of the same form as the equation referred to. We thus obtain the solutions ^ - P, (/i) . P, (0 (7) and ^ = P, (,*) . Q, (a (8) where Qn a) = Pn a) f f{i*n(C)}* ({*-!)' = iP, (0 log |i-J - ^^ P„_x (0 - ^^-P,_.(0-. . . . ni f ff,.^. . (n+l)(n + 2) 1^ ^ 2(2n + 3) ^ 1 . 3 . . . (2n + 1) (n+ 1) (n + 2) (n + 3) (n + 4) | ^ 2.4(2nV3)(2n + 5) " ^ +...|. .-W The solution (7) is finite when ^ = 1, and is therefore adapted to the space within an ellipsoid of revolution; whilst (8) is infinite for ^ = 1, but vanishes for £ = qo , and is therefore appropriate to the external region. As particular cases of the formula (9) we note e.(0 = iiog|^. ei(0 = inog[^J-i. Q» (0 = i (3C* - 1) log 1^ - K- The definite-integral form of Q„ shews that The corresponding expressions for the stream-function are readily found ; thus, from the definition of Art. 94, (11) whence |= - *(C* " D |f . | = M1-/^«)| (12) Thus, in the case of (7), we have <^-'^-¥-'-^i<'-'">'^'}' n (n + 1) * Fenren, c. t. ; Todhunter, o. vi. ; Fonyth, Arts. 9S-99. 104r-106] Motion of an Ovary Ellipsoid 136 "^«°«« ^ = M^) ^' - "'^'^^^ • ^^' - '^ ^ ^''^ The same result will follow of course from the second of equations (12). In the same way, the stream-function corresponding to (8) is 105. We can apply this to the case of an ovary ellipsoid moving parallel to its axis in an infinite mass of liquid. The elUptic co-ordinates must be chosen so that the ellipsoid in question is a member of the confocal family, say that for which ^ = Co* Comparing with Art. 103 (2) we see that if a, c be the polar and equatorial radii, and e the eccentricity of the meridian section, we must have The surface-condition is given by Art. 97 (1), viz. we must have - - \Uk^ (1 - /x«) (C« - 1) + const., (1) for C=» Co- Hence putting n= 1 in Art. 104 (14), and introducing an arbitrary multiplier -4, we have ^ = i^i(l-M»)({*-l)|ilog|3-}-^r^}, (2) with the condition (9) The corresponding formula for the velocity-potential is ^ = ^M{Klog|^-l[ (4) The kinetic energy, and thence the inertia-coefficient due to the fluid, may be readily calculated, if required, by the formula (5) of Art. 94. 106. Leaving the case of symmetry, the solutions of V*^ = when ^ is a tesseral or sectorial harmonic in /a and a> are found by a similar method to be of the types ^ = P,'(/*)..P«*U)3^)««, (1) ^ = P„' (f*) . e»' (0 3°^} «-. (2) where, as in Art. 86, P,*(/x) = (1 - ^*)** '^'^"f^^ (3) 136 Irrotational Motion of a Liquid [chap. V whilst (to avoid imaginaries) we write and g„.(i;) = ((;._i)i.*|«l^). It may be shewn that (4) (5) QO dt whence P,« (0 ^%^ - ^^^ Qn' (0 = (-)'+^ { {Pn' («}* . (C« - 1) ' • ■ (n + ») I 1 di dC (n-»)!i»-r ..(6) ..(7) As examples we may take the case of an ovary ellipsoid moving parallel to an equatorial axis, say that of y, or rotating about this axis. 1°. In the former case, the surface-condition is ioT ^ — ^, where F is the velocity of translation, or ?^=_ F. K - ..t^i «-— , . — ; ^ (1 — U*)' 008 ft» This is satisfied by putting n = 1, « = 1, in (2), viz. ^ = 4 (1 - M*)* (i«- 1)* . {i log 1^ - -^Fin} «» *-' the constant A being given by (8) ...(9) ^fi, ^0 + 1 ^o*-2 ) = -kV. (10) 2°. In the case of rotation about Oy, if fly be the angular velocity, we must have for f = Jo or ^ = k*n,y . j . M (1 — M*)*8in co. 35 (Co* - 1)* Putting n = 2, « « 1, in the formula (2) we find (11) ^ = ^/.(l~ ,.«)»(£«- l)*|fClog|^- 3 -^^jsin CO, ..(12) A being determined by comparison with (11). 106-107] Formulae far Planetary EUipBoid 137 107. When the ellipsoid is of the (Mode or 'planetary' form, the appropriate co-ordinates are given by a? s=r i cos sinh ly = i/xf, y = to cos co, z = m sin (a,^ ., . it V • • . . ^1 j ^««*« «/ — n,«x**^ v^*x.; — ,..VX — p y ( J* + l)^ J Here £ may range from to oo (or, in some applications from — oo through to + 00 ), whilst /A Ues between db 1. The quadrics ^ «= const., /i =» const, are planetary ellipsoids, and hyperboloids of revolution of one sheet, all having the common focal circle x = 0, g7 = A;. As Umiting forms we have the ellipsoid ^ = 0, which coincides with the portion of the plane a; == for which w <hy and the hyperboloid /a = coinciding with the remaining portion of this plane. With the same notation as before we find 8», = *(f-±^^,*) 8,., 8«f = A (^l±-^y 8f , 8«. = i(l-/*»)»(i:«+l)*8«,; (2) and the equation of continuity becomes This is of the same form as Art. 103 (4), with t^ in place of C> <^nd the like correspondence will run through the subsequent formulae. In the case of sjrmmetry about the axis we have the solutions ^ = p, (,*) . p, (a w and ^ = ^« (f*) . 3« (0. (5) . ,y. 1. 3.5 ... (2n- 1) where j)„ (0 = -j^ ^ ^ 2 (2n - 1) ^ n(n-l)(n-2)(n-3) 1 ^ 2.4(2n-l)(2n-3) ^ -l-.-.J, --W and . gn(0-y,(0/; ^^(^)^f[^,^i) . = ♦>' f r-,-1 _ (n+l)(n + 2) , 1.3.5 ... (2n+l) r 2(2n + 3) ^ (n + 1) (n + 2) (n + 3) (n + 4 ) .__, _ ) ,7. "^ 2.4(2n + 3)(2n + 6) ^ '**J 138 IrrotdtioncU' Motion of a Liquid [ohap. v the latter expansion being however convergent only when J > 1 *. As before, the solution (4) is appropriate to the region included within an ellipsoid of the family ^ = const., and (5) to the external space. Wenotethat p„ (0^^^ - ^^^^,(0 = - ^^^ (8) As particular cases of the formula (7) we have Jo (C) = cot-i t, ?i (C) = 1 - { cot-» C, ?.(a-i(3i*+l)cot-U-fC. The formulae for the stream-function corresponding to (4) and (6) are *-»-isTi)"-''''^'<^+'>%r^ '«' 108. V. The simplest case of Art. 107 (5) is when n = 0, viz. ^ = 24cot-iJ, (1) where ^ is supposed to range from — qo to + <3o . The formula (10) of the last Art. then assumes an indeterminate form, but we find by the method of Art. 104, ^^Akfi (2) This solution represents the flow of a liquid through a circular aperture in an infinite plane wall, viz. the aperture is the portion of the plane yz for which m <h. The velocity at any point of the aperture (J = 0) is IdJf A since, when a? = 0, i/x = (i* — tn*)*. The velocity is therefore infinite at the edge. Compare Art. 102, 3°. 2°. Again, the motion due to a planetary ellipsoid (J = Jo) moving with velocity TJ parallel to its axis in an infinite mass of Uquid is given by ^ = ^/i(l-Jcot-U), 0-i^*(l-/*»)(5*+l)j^-cot-n},...(3) where A = — hU -h- [tt^^ — cot"^ Jo[ • Denoting the polar and equatorial radii by a and c, and the eccentricity of the meridian section by 6, we have a = iJo, c = i (U + 1)*, e = (V + 1)-*. * The reader may easily adapt the demonstrations referred to in Art. 104 to the present case. 107-108] Streaan-Idnes of a Circular Disk In terms of these quantities il-- J7c4-|(l-e»)*-j8m-icl. . 139 ...(4) The forms of the lines of motion, for equidistant values of 0, are shewn below. Cf. Art. 71, 3°. X' X The most interesting case is that of the circular disk, for which 6 = 1, and A ^ 2Uc/Tr. The value of <f> given in (3) becomes equal to ± Afi, or ± -4 (1 — m^jt^y, for the two sides of the disk, and the normal velocity to ± U. Hence the formula (4) of Art. 44 gives 2T^^p<^U\ (5) as in Art. 102 (20). 140 Irrotcttional Motion of a Liquid [chap, y 109. The solutions of the equation Art. 107 (3) in tessecal harmonics are and ^ - P.* (^) . p/ (0 . ^1 to, (1) ^ = P„* (/t) . ?«• (C) . ^} to, (2) where j,,.(^) = (^« + l)f^lO, (3) and g,.(0 = (C»+l)*'^^, These functions possess the property ^ . a) ^n a) _ dp^'io / y+i(!L±iL' J_ (5) ^•^ ^^^ dC dC *•* ^^' ^ ' (n-*)!C«+r •••^^ We may apply these results as in Art. 108. 1^. For the motion of a planetary ellipsoid (^ = £o) parallel to' the axis of y we have n = 1, « = 1, and thence ^-4(l-,.«)*(J«+l)*|^^-cot-icJcosa>, (6) with the condition ^ — — ^ ^> for J = ^o> y denoting the velocity of the solid. This gives ^ld^-^°*"4=-*'^- (7) In the case of the disk (^q <= 0), we have ^ » 0, as we should expect. 2°. A^in, for a planetary eUipeoid rotating about the axis of y with angular velocity Oy, we have, putting n = 2, « = 1, ^ = ^/* (1 - m*)* a* + 1)* JSC cot-1 i - 3 + ^^1 sin «, . . .(8) with the surface-condition = *!0» . u (1 - u«)* sin a> (9) iU + 1)* 109-110] EUipsoidal Envelope 141 For the circular disk (f^ »= 0) this gives %7tA = - hm^ (10) At the two surfaces of the disk we have 4>^^2Ayi(l^ /i«)*sinco, 1^ = T Wl^ (1 - /i«)*sinco, -and substituting in the formula we obtain 2T = ^pc^ . il^* (11) 110. In questions relating to eUipeoids with three unequal axes we may employ the more general type of ElUpeoidal Harmonics, usually known by the name of * Lamp's Functions*.' Without attempting a formal account of these functions, we will investigate some solutions of the equation VV-=0, (1) in ellipsoidal co-ordinates, which are analogous to spherical harmonics of the first and second orders, with a view to their hydrodynamical appUcations. It is convenient to prefix an investigation of the motion of a liquid contained in an eUipeoidal envelope, which can be treated at once by Cartesian methods. Thus, when the envelope is in motion parallel to the axis of x with velocity U, the enclosed fluid moves as a solid, and the velocity-potential is simply <^ == — Ux. Next let us suppose that the envelope is rotating about a principal axis (say that of x) with angular velocity Q^^. The equation of the surface being X* V* z^ a* 6* c* ^ ' the surface-condition is a* dx 6« ay c« dz " 6« *^ ^ c« ^•^• We therefore assume <f> = Ayz, which is evidently a solution of (1), and obtain, on determining the constant by the condition just written, 6* — c* * See, for example, Ferrers, Spherical Harmonics, o. vL; W. D. Niven, PhiL Trans. A, t. clzxxii. (1801) and Proc, Roy. Soc. A, t. Ixxix. p. 458 (1906); Poincar^, Figures (T^quilibre d^une Masse Fluide, Paris, 1902, c. vi. ; Darwin, PhU Trans. A, t. cxovii p. 461 (1901) [ScienHfic Papers, Cambridge, 1907-11, t. ill. p. 186]. An outline of the theory is given by Wangerin, Lc. ante p. 103. 142 Irrotational Motion of a Liquid [chap, v Hence, if the centre be moving with a velocity whose components are TJy F, W and if !!«, fl^, fl, be the angular velocities about the principal axes, we have by superposition* 4> Vx-Vy-Wz-^^^n,yz-'p~n^zx-^^il,xy....{Z) We may also include the case where the envelope is changing its form as well as position, but so as to remain elUpsoidal. If the axes are changing at the rates d, 6, c, respectively, the general boundary-condition, Art. 10 (3), becomes g* + ^'^3* + Sl + ^| + S|-»- («) which is satisfied t by The equation (1) requires that ^ + - + !: = 0, 6) a b c which is in fact the condition which must be satisfied by the changing ellipsoidal surface in order that the enclosed volume (^abc) may be constant. 111. The solutions of the corresponding problems for an infinite mass of fluid bounded internally by an ellipsoid involve the use of a special system of orthogonal curvilinear co-ordinates. If X, y, z be functions of three parameters A, /a, v, such that the surfaces A = const., fi = const., v = const (1) are mutually orthogonal at their intersections, and if we write l,.(8£)V(|!y+(|!)', V \9/^/ V3ft/ \dfiJ »!.-©'-g)'-©'-J (2) * This result appears to have been published independently by Beltrami, Bjerknes, and Maxwell, in 1873. See Hicks, "Report on Recent Progress in Hydrodynamics," BriL Asa. Rep. 1882, and Kelvin's Papers, t. iv. p. 197 (footnote). t C. A. Bjerknes, "Verallgemeinerung des Problems von den Bewegungen, welohe in einer ruhenden unelastischen FlUssigkeit die Bewpgung eines Ellipsoids hervorbringt," OWinger Nachrichten, 1873. 110-112] Orthogonal Co-ordinates 143 the direction-cosines of the normals to the three surfaces which pass through (», y, z) will be paA' '^aX' *^aAJ' V^v *«v **8^j' l^'a;;' *»ai;' *»ci;J' •••<3) respectively. It follows that the lengths of linear elements drawn in the directions of these normals will be SA/Ai, Sft/Aj} Sv/Ag. Hence if <^ be the velocity-potential of any fluid motion, the total flux into the rectangular space included between the six surfaces A db |8A, ^db J8^, v± \hv will be It appears from Art. 42 (3) that the same flux is expressed by ^^<f> multiplied by the volume of the space, i.e. by SXhfiSv/hih^h^, Hence* ^■*-»A».g(j^J)^|.(i|)4(4|*))...(4) Equating this to zero, we obtain the general equation of continuity in orthogonal co-ordinates, of which particular cases have already been investi- gated in Arts. 83, 103, 108. The theory of triple orthogonal systems of surfaces is very attractive mathematically, and abounds in interesting and elegant formulae. We may note that if A, ft, v be regarded as fimctions of x, y, z, the direction-cosines of the three line-elements above considered can also be expressed in the forms [h^dx' h^dy' h^dzj' [h^dx' h^dy' h^dzj' [h^dx' h^dy' h^dzj' (5) from which, and from (3), various interesting relations can be inferred. The formulae already given are, however, sufficient for our present purpose. 112. In the applications to which we now proceed the triple orthogonal system consists of the conf ocal quadrics • "* +r/\+.-^.-l = 0. (1) a^-\-e b^ + e c^ + d * The above method was given in a paper by W. Thomson, *'0n the Equations of Motion of Heat referred to Curvilinear Co-ordinates/' Camb. Math. Joum. t. iv. (1843) [Papers, t. i. p. 25]. Reference may also be made to Jacobi, "Ueber ^ine particul&re Losung der partiellen Diffe- rentialgleiohung ," CreUe, t. xxxvi. (1847) [Werke, t. ii. p. 198]. The transformation of V*0 to general orthogonal co-ordinates was first effected by Lam^, '*Sur les lois de T^quilibre du fluide ^th^r6," Joum. de T£c6U Polyi, t. ziv. (1834). See also Lefons 9ur les Coordonndea Curvilignee, Paris, 1869, p. 22. 144 Irrotational Motion of a Idquid [OHAP. V whose properties are explained in books on Solid Geometry. Through any given point (x, y, z) there pass three surfaces of the system, corresponding to the three roots of (1), considered as a cubic in 0. If (as we shall for the most part suppose) a > 6 > c, one of these roots (A, say) will lie between qo and — c\ another (/x) between — c* and — 6*, and the third (v) between — 6* and — a*. The surfaces A, ft, v are therefore ellipsoids, hyperboloids of one sheet, and hyperboloids of two sheets, respectively. It follows immediately from this definition of A, /a, v, that x^ + .7* + _^ (\^d)(ii^d)(v-d) 5« + (a« + e) (6« + 0) (c« + <?)'•• • '^^^ a« + d ^ 6« + c identically, for all values of 6. Hence multiplying by a* + d, and afterwards putting = — a^y we obtain the first of the following equations : x^ y' ««« (a« + A) ( g g + fi) (g' + v) (g« - 6«) (g2 - c2) (6« + A) (6« + ji) (6« + i;) (6* - c«) (6« - g«) (c« + A) (c« 4- /x) (c2 + v) (c« - g«) (c« - 6«) (3) These give Sx _ . X ^_ i y a»+A' aA 2 6« + A' aA~'3*TA' 92 _ J z (4) and thence, in the notation of Art. Ill (2), + y* :.+ A)* ^ {b* + A)» "^ (c» + A)«J (5) If we differentiate (2) with respect to 6 and afterwards put fl == A, we deduce the first of the following three relations : i.._. («' + ^)(ft* + A)(c« + A) ] (A-^)(A-v) • A « = 4 («* + m) (ft* + m) (c* + li) (m - v) (/*'- A) ' ;i 2 = 4 (o' + v) (6* + v) {c* +v) * (v-A)(v-/t) • (6) The remaining relations of the sets (3) and (6) have been written down from symmetry*. • It will be noticed that *j. A,, A, are double the perpendioolan from the origin on the tangent planes to the three quadrios X, n, r. 112-113] Orthogonal Co-ordinates 145 Substituting in Art. Ill (4), we find* + (v - A) |(a« + ,*)* (6« + ^)» (c« + ,*)» 1^1* + (A - /*) |(o« + v)* (6* + v)* (c» + v)* l^ri 4>. (7) 113. The particular solutions of the transformed equation V^ = which first present themselves are those in which <^ is a function of one (only) of the variables A, /a, v. Thus ^ may be a function of A alone, provided (a« + A)* (62 + A)* (c* + A)*^ = const., whence <f> = C j -r-, (1) if A = {(a« + A) (6« + A) (c« + A)}*, (2) the additive constant which attaches to <f> being chosen so as to make <f> vanish for A = oo . In this solution, which corresponds to ^ = A/r in spherical harmonics, the equipotential surfaces are the confocal ellipsoids, and the motion in the space external to any one of these (say that for which A = 0) is that due to a certain arrangement of simple sources over it. The velocity at any point is given by the formula -».S-4 <») At a great distance from the origin the ellipsoids A become spheres of radius A*, and the velocity is therefore idtimately equal to 2C/r*, where r denotes the distance from the origin. Over any particular equipotential surface A, the velocity varies as the perpendicular from the centre on the tangent plane. To find the distribution of sources over the surface A = which would produce the actual motion in the external space, we substitute for <f> the value (1), in the formula (11) of Art. 58, and for <f)' (which refers to the internal space) the constant value •*dA ^'-af w * Cf. Lam^, "Sur les surfaces itothermeB dans les corps solides homog^nes en ^quilibre de temperature," LiouviOe, t. u. (1837). L. H. 10 146 Irrotational Motion of a Liquid [chap, v The formula referred to then gives, for the surface-density of the required distribution, ic-^' (^) The solution (1) may also be interpreted as representing the motion due to a change in the dimensions of the ellipsoid, such that the surface remains similar to itself, and retains the directions of its principal axes unchanged. If we put d/a = b/b = c/c, = A, say, the surface-condition Art. 110 (4) becomes — d<f>/dn = ^khi, which is identical with (3), if we put C = Ikdbc. A particular case of (5) is where the sources are distributed over the elliptic disk for which A = — c^, and therefore z* = 0. This is important in Electrostatics, but a more interesting application from the present point of view is to the flow through an elliptic aperture, viz. if the plane ay be occupied by a thin rigid partition with the exception of the part included by the ellipse ^2 + p = i' « = 0' we have, putting c = in the previous formulae, (6) ■/ (a* + A)* (6« + A)* A* ' where the upper limit is the positive root of ""^ +./^^ + T = l, (7) a* 4- A 62 4- A A and the negative or the positive sign is to be taken according as the point for which <f> is required lies on the positive or the negative side of the plane a>y. The two values of <f> are continuous at the aperture, where A = 0. As before, the velocity at a great distance is equal to 2^/r*, nearly. For points in the aperture the velocity may be found immediately from (6) and (7) ; thus we may put Sn .....(x-g-g)', 8, = -!^* approximately, since A is small, whence This becomes infinite, as we should expect, at the edge. The particular case of a circular aperture has already been solved otherwise in Arts. 102, 108. 113-114] Translation of an Ellipsoid 147 114. We proceed to investigate the solution of V*^ = 0, finite at infinity, which corresponds, for the space external to the elUpsoid, to the solution <f> = X ioT the internal space. Following the analogy of spherical harmonics we may assume for trial <f> = ^y (1) which gives V^X + - 5^ = 0, xdx (2) and inquire whether this can be satisfied by making x equal to some function of A only. On this supposition we shall have, by Art. Ill, dx'^Hy^^dr and therefore, by Art. 112 (4), (6), xdx (A — /i) (A — v) dX' On substituting the value of V*;^ in terms of A, the equation (2) becomes |(a« + A)* (6« + A)» (c« + A)* ^\=- (6* + A) (c* + A) ^, which may be written ^,log{(a3+A)*(63 + A)t(c^ + A)*|| = -^,. Hence X^oT . -, ., (3) J M«^ + A)* (6* + A)t (c« + A)* the arbitrary constant which presents itself in the second integration being chosen as before so as to make x vanish at infinity. The solution contained in (1) and (3) enables us to find the motion of a liquid, at rest at infinity, produced by the translation of a solid ellipsoid through it, parallel to a principal axis. The notation being as before, and the ellipsoid o« ^ 6« ^ c» + Si + !i=l (4) being supposed in motion parallel to x with velocity U, the surface- condition is |=-17|. forA = (5) Let us write, for shortness, (6) 10—2 148 Irrotational Motion of a Liquid [chap, v where A = {(a^ + A) (6* + A) (c^ + A)}* ; (7) it will be noticed that these quantities <io, jSq, y© *re purely numerical. The conditions of our problem are satisfied by * = C./;p^. (8) provided C = ^— — U (9) The corresponding solution when the ellipsoid moves parallel to y or 2; can be written down from symmetry, and by superposition we derive the case where the ellipsoid has any motion of translation whatever*. At a great distance from the origin, the formula (8) becomes equivalent to <A = *c^, (10) which is the velocity-potential of a double source at the origin, of strength §7rO, or compare Art. 92. m The kinetic energy of the fluid is given by where I is the cosine of the angle which the normal to the surface makes with the axis of x. Since the latter integral is equal to the volume of the elUpsoid, we have 2T = -^r^^— . ^nabcp . U^ (11) The inertia-coefl&cient is therefore equal to the fraction 0^/(2 — Oq) of the mass displaced by the soUd. For the case of the sphere (a = 6 = c) we find oo = |; this makes the fraction equal to i, in agreement with Art. 92. If we put 6 = c, we get the case of an ellipsoid of revolution, including (for a = 0) that of a circular disk. The identification with the results obtained by the methods of Arts. 105, 106, 108, 109 for these cases may be left to the reader. * This problem was first solved by Green, "Researches on the Vibration of Pendulams in Fluid Media " Trans. B. S. Edin. 1833 [Papers, p. 315]. The investigation is much shortened if we assume at once from the Theory of Attractions that (8) is a solution of V*0=O, being in fact (except for a constant factor) the :r-component of the attraction of a homogeneous ellipsoid at an external point. 114-115] Rotation of an EUipaoid 149 115. We next inquire whether the equation V*^ = can be satisfied by 4> = y^Xy (1) where x is » function of A only. This requires ^--^itAi-"- (2) Now, from Art. 112 (4), (6), ydy zdz ^ \y 3A z dX) dX _ (o« + A) (6« + A) (c' + A) / 1 1 \dx (A - u) (A - v) U* + A "*" c» + A/ dX' (A-/*){A-v) On substitutioQ in (2) we find, bj Ait. 112 (7), ^, log {(«« + A)» (6. + A)» (c. + A)* 1} = - ^ - 3^. whence X° gj, (fc. + A) (o« + A) A ^^^ the second constant of integration being chosen as before. For a rigid ellipsoid rotating about the axis of x with angular velocity Qajj the surface-condition is I-°.(»|-»IS- •• <*) for A = 0. Assuming* dX <f> = Cyzf ./* (6» + A) {c« + A) A ^^^ we find that the sniface-condition (4) is satisfied, provided The formulae for the cases of rotation about y ot z can be written down from symmetry f. * The expression (5) differs only by a factor from where 4 is the gravitation-potential of a nniform solid ellipsoid at an external point (a;, y^ 2;). Since V*<^=0 it easily follows that the above is also a solution of the equation VV=0. t The solution contained in (5) and (6) is due to Qebsch, "Ueber die Bewegung eines EOipsoidee in einer tropfbaren Flussigkeit," Crdit, tt. lii. liii. (1866-7). 150 Irrotational Motion of a Liquid [chap, v The formula for the kinetic energy is if ({, m, n) denote the direction-cosines of the normal to the ellipsoid. The latter integral = Hi (y' - 2*) dxdydz = H&* - '«') • l^«^. Hence we find The two remaining types of ellipsoidal harmonic of the second order, finite at the origin, are given by the expression x" y* 2* a*+^ h^ + B c^-^B -1 (8) where B is either root of •- :;i + r* - >» + -? /» =0 (9) this being the condition that (8) should satisfy v*^ =0. The method of obtaining the corresponding solutions for the external space is explained in the treatise of Ferrers. These solutions would enable us to express the motion produced in a surrounding liquid by variations in the lengths of the axes of an ellipsoid, subject to the condition of no variation of volume : dja -^b/b +c/c=0 (10) We have already found, in Art. 113, the solution for the case where the eUipsoid expands (or contracts) remaining similar to itself; so that by superposition we could obtain the case of an internal boundary changing its position and dimensions in any manner what- ever, subject only to the condition of remaining eUipsoidal. This extension of the results arrived at by Green and Clebsch was first treated, though in a different manner from that here indicated, by Bjerknes*. 116. The investigations of this chapter have related almost entirely to the case of spherical or ellipsoidal boundaries. It will be understood that solutions of the equation V^ = can be carried out, on lines more or less similar, which are appropriate to other forms of boundary. The surface which comes next in interest, from the point of view of the present subject, is that of the anchor-ring, or 'torus'; this case has been very ably treated, by distinct methods, by Hicks, and Dyson f. We may also refer to the analytically remarkable problem of the spherical bowl, which has been investigated by Basset J. * Le. ante p. 142. t Hicks, "On Toroidal Functiona," Phil. Trans. 1881; Dyaon, "On the Potential of an Anchor-Ring," PhU. Trans. 1893 ; see also 0. Neumann, Ic. ante p. 127. X "On the Potential of an Electrified Spherical Bowl, &c.," Proc. Lond. Math. 8oc. t. zvi. (1885); Hydrodynamics, t. i. p. 149. CHAPTER VI ON THE MOTION OF SOLIDS THROUGH A LIQUID : DYNAMICAL THEORY 117. In this chapter it is proposed to study the very interesting dynamical problem furnished by the motion of one or more solids in a frictionless liquid. The development of this subject is due mainly to Thomson and Tait* and to Kirch hofff* 1*^® cardinal feature of the methods followed by these writers consists in this, that the solids and the fluid are treated as forming one dynamical system, and thus the troublesome calculation of the effect of the fluid pressures on the surfaces of the soUds is avoided. To begin with the case of a single solid moving through an infinite mass of liquid, we will suppose in the first instance that the motion of the fluid is entirely due to that of the solid, and is therefore irrotational and acyclic. Some special cases of this problem have been treated incidentally in the foregoing pages, and it appeared that the whole effect of the fluid might be represented by an addition to the inertia of the solid. The same result will be found to hold in general, provided we use the term 'inertia' in a somewhat extended sense. Under the circumstances supposed, the motion of the fluid is characterized by the existence of a single-valued velocity-potential ^ which, besides satisfying the equation of continuity vv = o, (1) fulfils the following conditions : (1°) the value of — 3^/3n, where 8n denotes as usual an element of the normal at any point of the surface of the solid, drawn on the side of the fluid, must be equal to the velocity of the surface at that point normal to itself, and (2°) the differential coefficients d<f>/dx, i * Natural Philosophy, Art. 320. Subsequent inveetigatioiiB by Lord Kelvin will be referred to later. t "Ueber die Bewegnng eines Rotationakorpers in einer Fliissigkeit/' CreUe, t. Ixzi. (1869) [Qea, AhK p. 376]; Mechanik, c. ziz. 152 Motion of Solids through a Liquid [chap, vi d<f>ldy, d<f>ldz must vanish at an infinite distance, in every direction, from the solid. The latter condition is rendered necessary by the consideration that a finite velocity at infinity would imply an infinite kinetic energy, which could not be generated by finite forces acting for a finite time on the solid. It is also the condition to which we are led by supposing the fluid to be enclosed within a fixed vessel infinitely large and infinitely distant, all round, from the moving body. Foi.' on this supposition the space occupied by the fluid may be conceived as made up of tubes of flow which begin and end on the surface of the solid, so that the total flux across any area, finite or infinite, drawn in the fluid must be finite, and therefore the velocity at infinity zero. It has been shewn in Art. 41 that under the above conditions the motion of the fluid is determinate. 118. In the further study of the problem it is convenient to follow the method introduced by Euler in the dynamics of rigid bodies, and to adopt a system of rectangular axes Ox, Oy, Oz fixed in the body, and moving with it. If the motion of the body at any instant be defined by the angular velocities p, J, r about, and the translational velocities w, v, w of the origin parallel to, the instantaneous positions of these axes*, we may write, after Kirchhoff, <l> = u<f>i + v<f>^-hw<f>^ + pXi + qXi + rxsy (2) where, as will appear immediately, ^i, ^2» ^3> Xi> Xa* Xs *^® certain functions of X, y, z determined solely by the configuration of the surface of the solid, relative to the co-ordinate axes. In fact, if I, m, n denote the direction-cosines of the normal, drawn towards the fluid, at any point of this surface, the kinematical surface-condition is ■ — ^ = l{u -{- qz — ry) + m{v -{- rx — pz) + n{w -{- py — qx), whence, substituting the value (2) of <f>, we find dK -^' "a^T -^' "- =^ ^^^^ny-mz. --^ = lz-nx, ^^=.mx^ly. \o) Since these functions must also satisfy (1), and have their derivatives zero at infinity, they are completely determinate, by Art. 41 f . * The symbols u, v, tc, p, q, r are not at present required in their former meanings, f For the particular ease of an ellipsoidal surface, their values may be written down from the results of Arts. 114, 116. 117-119] ImpuUe of the Motion 153 119. Now whatever the motion of the solid and fluid at any instant, it might have been generated instantaneously from rest by a properly adjusted impulsive 'wrench' applied to the soUd. This wrench is in fact that which would be required to counteract the impulsive pressures p<f> on the surface, and, in addition, to generate the actual momentum of the solid. It is called by Lord Kelvin the 'impulse' of the system at the moment under con- sideration. It is to be noted that the impulse, as thus defined, cannot be asserted to be equivalent to the total momentum of the system, which is indeed in the present problem indeterminate*. We proceed to shew however that the impulse varies, in consequence of extraneous forces acting on the solid, in exactly the same way as the momentum of a finite dynamical system. Let us in the first instance consider any actual motion of a solid, from time to to time ^^9 under any given forces applied to it, in a jmite mass of liquid enclosed by a fixed envelope of any form. Let us imagine the motion to have been generated from rest, previously to the time ^, by forces (whether continuous or impulsive) applied to the solid, and to be arrested, in like manner, by forces applied to the solid after the time t^. Since the momentum of the system is null both at the beginning and at the end of this process, the time-integrals of the forces appUed to the solid, together with the time-integral of the pressures exerted on the fluid by the envelope, must form an equilibrating system. The effect of these latter pressures may be calculated, by Art. 20, from the formula f = |-k' + y(0 (1) A pressure uniform over the envelope has no resultant effect ; hence, since <f> is constant at the beginning and end, the only effective part of the integral pressure Ipdtis given by the term -\pii^^ (2) Let us now revert to the original form of our problem, and suppose the containing envelope to be infinitely large, and infinitely distant in every direction from the moving solid. It is easily seen by considering the arrangement of the tubes of flow (Art. 36) that the fluid velocity j at a great distance r from an origin in the neighbourhood of the solid will ultimately be, at mostf, of the order 1/r*, and the integral pressure (2) therefore of the order 1/r*. Since the surface-elements of the envelope are of the order r^Stu, where hm is an elementary sohd angle, the force- and couple-resultants of the integral pressure (2) will now both be null. The same statement * That is, the attempt to calculate it leads to 'improper* or 'indeterminate' integralB. t It is really of the order l/r* when, as in the case considered, the total flux outwards is zero. 154 Motion of Solids through a Liquid [chap, vi therefore holds with regard to the time-integral of the forces applied to the solid. If we imagine the motion to have been started insUvnianeously at time ^0, and to be arrested instantaneously at time ^i, the result at which we have arrived may be stated as follows : The 'impulse' of the motion (in Lord Kelvin's sense) at time i^ differs from the 'impulse' at time ^o l>y the time-integral of the extraneous forces acting on the solid during the interval i^ — ^q*. It will be noticed that the above reasoning is substantially unaltered when the single solid is replaced by a group of soUds, which may moreover be flexible instead of rigid, and even when these solids are replaced by masses of Uquid which are moving rotationally. 120. To express the above result analytically, let £, ly, J, A, /i, v be the components of the force- and couple-constituents of the impulse; and let X^ y, Z, i, Jf , iV designate in the same manner the system of extraneous forces. The whole variation of |, ry, ^, A, /i, i/, due partly to the motion of the axes to which these quantities are referred, and partly to the action of the extraneous forces, is then given by the formulae f = uyq ^ vQ + rfjL - qi^ -{- Ly ^=u^--wi + pu^rX + M,\ (1) — = V^ - 7/7^ -f. grA - ;?/i 4- N. For at time t ■■{- &t the moving axes make with their positions at time t angles whose cosines are (1, rSt, - qSt), (- rSt, 1, pSt), (gr&, - p8t, 1), respectively. Hence, resolving parallel to the new position of the axis of x, ^ + Si=i + rj.rSt-^.q8t + XBL Again, taking moments about the new position of Ox, and remembering that has been displaced through spaces uSt, vSt, w8t parallel to the axes, we find A + 8A = A + t; . t^8« - ^ . vSt + /i . r8« - V . gr8« + LSt. These, with the similar results which can be written down from symmetry, give the equations (1). ♦ Sir W. Thomson, l.c. anie p. 31. The form of the argument given above was kindly suggested to the author by Sir J. Larmor. t Cf. Hayward, "On a Direct Method of Estimating Velocities, Accelerations, and all similai Quantities, with respect to Axes moveable in any manner in space,*' Camb, Trana. t. X. (1866). di' = r.j- -qUX, dri di' -vl- -ri+Y, dt dt' = qi- -pq+Z, 119-121] Kinetic Energy 155 When no extraneous forces act, we verify at once that these equations have the integrals f* + 17* + i* = const., X^ -\- ii-q -\- vl = const., (2) which express that the magnitudes of the force- and couple-resultants of the impulse are constant. 121. It remains to express |, -q, J, A, /i, 1/ in terms of w, v, w^ p, j, f. In the first place let T denote the kinetic energy of th^fiutd, so that 2T —Mti^. m where the integration extends over the surface of the moving solid. Substituting the value of <f> from Art. 118 (2), we get 2T = Aw« + Bv« + Gw^ + 2A't;u? + 2B'wu + 2G'uv + Py« + Qj« + Rr* + 2P'3T + 2QVp + 2R'j)g + 2p (Pw + Ov + Hw) + 2q (T'u + G'v + K'w) + 2r (P''w + Q'^v + B."w), (2) where the 21 coefficients A, B, C, &c. are certain constants determined by the form and position of the surface relative to the co-ordinate axes. Thus, for example, * — ,jj^^-iS.-,IJ^,^'^ \ (3) = " '' //^» fe' '^ = '' Ih' ("^ ~ "'''^ '^^• / the transformations depending on Art. 118 (3) and on a particular case of Green's Theorem (Art. 44 (2)). These expressions for the coefficients were given by Elirchhoff. The actual values of the coefficients in the expression for 2T have been found in the preceding chapter for the case of the ellipsoid, viz. we have from Arts. 114, 116 A— 5L. Uaabc p 1 (fe*-C)' ( y.-g.) . ^irpabc. (*) 156 Motion of Solids through a Liquid [chap, vi with similar expressions for B, 0, Q, B. The remaining coefficients, as will appear pre- sently, in this case all vanish. We note that ^-»=(2^Sy#k)-*''^' <') 80 that if a > 6 > c, then A < B < C, as might have been anticipated. The formulae for an ellipsoid of revolution may be deduced by putting 6 =c ; they may also be obtained independently by the method of Arts. 104-109. Thus for a circular disk (a =0, 6 =c) we have A,B, C=ipc»,0,0; P,Q, B=0, H/'Cfi.ifpc*. (6) The kinetic energy, .T^ say, of the solid alone is given by an expression of the form 2Ti =ft m (fi« -f t;2 -f w^) + PiP" + Qi?" + Rir* + 2PiV + 2Qi'rp + 2Ri'^ + 2in{a {vr — wq) + jS {wp — ur) -{■ y {uq — vp)} (7) Hence the total energy T + T, , of the system, which we shall denote by T, is given by an expression of the same general form as (2), say 2r = Au^ + Bv^ + Cw^ + 2A'vw + 2B'wu + 2C'uv -f Pf + Qq^ + JRr" + 2P'gr + 2QWp + 2Rjq -V2p{Fu-\-Gv + Hw) + 2gr {F'u + G'v -f H'w) + 2r {F''u + G"v + H"w), _ (8) where the coefficients are printed in uniform type, although six of them have of course the same values as in (2) 122. The values of the several components of the impulse in terms of the velocities u, v, w, p, q, r can now be found by a well-known dynamical method*. Let a system of indefinitely great forces (X, Y, Z, i, Jf, N) act for an indefinitely short time r on the soUd, so as to change the impulse from (L V> J» A, /i, v) to (f + 8|, ly + &7, J + S^ A + SA, /* + S/i, V + Si/). The work done by the force X, viz. /, Xudly Ues between Ui\ Xdt and w, I Xdt, Jo Jo where Ui and tig ^^^ ^^^ greatest and least values of u during the time r, i.e. it lies between w^S^ and WgSf . If we now introduce the supposition that 8|, S?y, S^, SA, 8/Lt, 8v are infinitely small, Wj and w, are each equal to u, and the work done is wSf . In the same way we may calculate the work done by * See Thomson and Tait, Art-. 313, or Maxwell, Electricity and Magnetism, Part. iv. c. v. 121-122] Relations betweefi Energy and Impvlse 157 the remaming forces and couples. The total result must be equal to the increment of the kinetic energy, whence w8f + vSry + t^SJ + p8A + 58/x + rSv Now if the velocities be all altered in any given ratio, the impulses wiU be altered in the same ratio. If then we take 8w _ Sv __ Si^ _ 8p _ 8} _ 8r _ , u ^ V " w ~ p q r ' it WiU follow that ^ = h = % ^^4 = ^-^ = - - ^■ Substituting in (1), we find ui + vr) + w^ + pX+ qfi + rv dT , dT , ar , dT ^ ar , ar „„ ,.,, = "a^ + ''a^ + «'a^ + Pa^ + «W"^''3^° ^' ••"^^^ since T is a homogeneous quadratic function. Now performing the arbitrary variation 8 on the first and last members of (2), and omitting terms which cancel by (1), we find Since the variations 8u, 8t;, iw^ 8p, Sq, 8r are all independent, this gives the required formulae ^' ''^ ^ "" du' dv' dw' ^' ^'"''^ dp' dq' dr ^"^^ It may be noted that since f , t?, J, ... are linear functions of w, v, «?, . . . , the latter quantities may also be expressed as linear fimctions of the former, and thence T may be regarded as a homogeneous quadratic function of £> Vy i> ^> /*> *'• When expressed in this manner we may denote it by T\ The equation (1) then gives at once u8f + vS?y + wS^ + pSX 4- qSfjL 4- rSv = -af^^ + ^^ + -ar^^ + 'aA^^+a^^'* + -air^''' , ar dr dr dr ar dr whence u, v, w = .^, -^. j^ , p, q, r = -^j, ^-, ^ (4) These formulae are in a sense reciprocal to (3). We can utilize this last result to obtain, when no extraneous forces act^ 158 Motion of Solids through a Liquid [chap, vi another integral of the equations of motion, in addition to those found in Art. 120. Thus dt 'di (ft "^ • • • "^ • • • + dX (ft "^ • • • "^ • • • _ df ^ ^ dX^ -t^^-t- ... + ... +?^+ ... + ..., which vanishes identically, by Art. 120 (1). Hence we have the equation of energy T = const (5) 128. If in the formulae (3) we put, in the notation of Art. 121, T = T + Ti, it is known from the dynamics of rigid bodies that the terms in T^ represent the linear and angular momentum of the solid by itself. Hence the remaining terms, involving T, must represent the system of impulsive pressures exerted by the surface of the soUd on the fluid, in the supposed instantaneous generation of the motion from rest. This is easily verified. For example, the x-component of the above system of impulsive pressures is = At^ + O'v + B*w + Pp + T'q -f P'V = gi, ... .(6) by the formulae of Arts. 118, 121. In the same way, the moment of the impulsive pressures about Ox is JJp^ (ny-mz)dS=-p fj<f> ^ dS 3T =^Fu-\-aV'{-B.w + 'Pp+Ii'q-i-Q'r= '^ (7) dp' 124. The equations of motion may now be written* d_dT^^dT_ 92; dtdu dv ^ dw ' dt dv ^dw du ' \ d dT dT dT ji* a-« "" ? a-. P Oi, "^ ^» (ft dw du dv dar_ dT^^^dT^ ^dT _ 92; dt dp "" dv dw dq ^ dr ' d dT dT dT , dT dT ^ ,, > (1) dt dq d dT dw du dT dr dp ^ dT _ _ dtdr du dv ^ dT 32"^ V f * See Kirchhoff, le. ante p. 151 ; also Sir W. Thomson, "Hydrokinetio Solations and Obser- vations," PhU, Mag. Nov. 1871 [reprinted in BaUimare Lectures, Cambridge, 1904, p. 684]. 122-124] Equations of Motion 159 If in these we write T = T + Ti, and isolate the terms due to T, we obtain expressions for the forces exerted on the moving solid by the pressure of the surroimding fluid ; thus the total component (X, say) of the fluid pressure parallel to x is and the moment (L) of the same pressures about cr is* L - - - — -I- 9T _ ar aT _ 8T dtdp dv dw dq ^ dr For example, if the solid be constrained to move with a constant velocity {u, V, w), without rotation, we have X, Y, Z = 0, ~ ~ ~ \ T *, ^T 3T aT aT aT bt bt >....(4) dv dw* dw du* du dv* where 2T = Au^ + Bv* + Cw^ + 2A'vw + 2B'wu + 2C'uv. The fluid pressures thus reduce to a couple, which moreover vanishes if aT aT aT du dv dw i.e. provided the velocity (w, v, w) be in the direction of one of the principal axes of the eUipsoid Ax^ + By* + Cz^ + 2A'yz + 2B'zx + 2C'xy = const (5) Hence, as was first pointed out by Kirchhoff, there are, for any solid, three mutually perpendicular directions of permanent translation; that is to say, if the soUd be set in motion parallel to one of these, without rotation, and left to itself, it will continue to move in this manner. It is evident that these directions are determined solely by the configuration of the surface of the body. It must be observed however that the impulse necessary to produce one of these permanent translations does not in general reduce to a single force; thus if the axes of co-ordinates be chosen, for simplicity, parallel to the three directions in question, so that A\ JB', C = 0, we have, corresponding to the motion u alone, i, 7), J = Au, 0, 0; A, /i, 1/ == Fu, F% F"u, 80 that the impulse consists of a wrench of pitch FjA. * The forms of these expressions being known, it is not difficult to verify them by direct calculation from the pressure-equation. Art. 20 (4). See a paper " On the Forces experienced by a Solid moving through a liquid,** Quart. Jowm. MtUk. t. xix. (1883). 160 Motion of Solids through a Liquid [chap, vi With the same choice of axes, the components of the couple which is the equivalent of the fluid pressures on the solid, in the case of any uniform translation (Uy v, w), are L, M, N = (B - O) vw, (O - A) vm, (A - B) w (6) Hence if in the ellipsoid kx^ + By« + Cz^ = const., (7) we draw a radius vector r in the direction of the velocity (w, v, w) and erect the perpendicular h from the centre on the tangent plane at the extremity of r, the plane of the couple is that of h and r , its magnitude is proportional to sin {h, r)/h, and its tendency is to turn the solid in the direction from h to r. Thus if the direction of (i/, v, w) differs but slightly from that of the axis of X, the tendency of the couple is to diminish the deviation when A is the greatest, and to increase it when A is the least, of the three quantities A, B, C, whilst if A is intermediate to B and C the tendency depends on the position of r relative to the circular sections of the above elUpsoid. It appears then that of the three permanent translations one only is thoroughly stable, viz. that corresponding to the greatest of the three coefficients A, B, C. For example, the only stable direction of translation of an elUpsoid is that of its least axis; see Art. 121*. 125. The above, although the simplest, are not the only steady motions of which the body is capable, under the action of no extraneous forces. The instantaneous motion of the body at any instant consists, by a well-known theorem of Kinematics, of a twist about a certain screw ; and the condition that this motion should be permanent is that it should not affect the configuration of the impulse (which is fixed in space) relatively to the body. This requires that the axes of the screw and of the corresponding impulsive wrench should coincide. Since the general equations of a straight line involve four independent constants, this gives four linear relations to be satisfied by the five ratios u : v : w : p : q : r. There exists then for every body, under the circumstances here considered, a singly-infinite system of possible steady motions. The steady motionfi next in importance to the three permanent translations are those in which the impulse reduces to a couple. The equations (1) of Art. 120 shew that we may have $, 7, f =0, and X, /x, v constant, provided \/p =fi/q = v/ry =1% say (1) If the axes of co-ordinates have the special directions referred to in the preceding Art., the conditions f, 17, f =0 give us at once u, v, to in terms of p, q, r, viz. Fp+rq + F'r, Qp+Q'q + G'r Hp+H'q+H''r ,., u^-^ ^ , t;= — ^ ^ , w= ^ ^ (2) * The physical cause of this tendency of a flat-shaped body to set itself broadside-on to the relative motion is clearly indicated in the diagram on p. 81. A number of interesting practical illustrations arc given by Thomson and Tait, Art. 325. 124--125] Steady Motions 161 Substituting these values in the expressions for X, thv obtained from Art. 122 (3), we find ae de de ^^^""^Tp' 8^' ¥ <^) provided 2e (p, q, r) = 9p^-\-^ +Kr« +2Wgr +24®'rp +2Vi'pq, (4) the coefficients in this expression being determined by formulae of the types ^ p F* G* H* ^, j^ F'F' Q'Q" WW ... These formulae hold for any case in which the force-constituent of the impulse is zero. Introducing the conditions (1) of steady motion, the ratios piqir are to be determined from the three equations 9p +Vi'q ■>r&T =lcp, K>+€g +Wr = kq, (6) &P+Wq+Vit = kr. The form of these shews that the line whose direction-ratios are piqir must be parallel to one of the principal axes of the ellipsoid e (ar, y, z) =const (7) There are therefore three permanent screw-motions such that the corresponding impulsive wrench in each case reduces to a couple only. The axes of these three screws are mutually at right angles, but do not in general intersect. It may now be shewn that in all cases where the impulse reduces to a couple only, the motion can be completely determined. It is convenient, retaining the same directions of the axes, to change the origin. Now the origin may be transferred to any point (a;, y, z) by writing u+ry-qz, v+pz-rx, to+qx-py, for u, V, w respectively. The coefficient of 2rr in the expression for the kinetic energy, Art. 121 (8), becomes -Bx+CT, that of 2wq becomes Cx +H\ and so on. Hence if we take ^=lU-c> ^-^c-a)' '=Ka'b) W the coefficients in the transformed expression for 2T will satisfy the relations B~ C ' C A' A^B ^^^ If we denote the values of these pairs of equal quantities by a, /9, y respectively, the formulae (2) may be written IMt — — ^ "X , V -— "~ *s f V/ — ^ jj^ , .......a.....'. ..^ ^^} where 2* (p, q, r) =2 J>* + ;b ?* '^'C^ +2aqr+2prp +2y pq (11) The motion of the body at any instant may be conceived as made up of two parts ; viz. a motion of translation equal to that of the origin, and one of rotation about an instantaneous axis paiMing through the origin. Since (, 7, ^=0 the latter part is to be determined by the equations ^=rM-gv, -^^P'-r^ 5=?X-PM. L. H. II 162 Motion of Solids through a Liquid [chap, vi which express that the vector (X» /a, y) is constant in magnitude and has a fixed direction in space. Substituting from (3), (12) d 9e_ ae_ ae di 9g "^ ¥ '" 3p' d a9_ 9g_ ae (ft ar^^ap ^ag' These are identical in form with the equations of motion of a rigid body about a fixed point, so that we may make use of Poinsot's well-known solution of the latter problem. The angular motion of the body is obtained by making the ellipsoid (7), which is fixed in the body, roll on a plane Xa; + /jty + v2 = coast. , which is fixed in space, with an angular velocity proportional to the length 01 of the radius vector drawn from the origin to the point of contact /. The representation of the actual motion is then completed by impressing on the whole system of rolling ellipsoid and plane a velocity of translation whose components are given by (10). This velocity is in the direction of the normal OM to the tangent plane of the quadric * (a:, y, 2) = -«», (13) at the point P where 01 meets it, and is equal to 0P,0M angular velocity of body (14) When 01 does not meet the quadric (13), but the conjugate quadric obtained by changing the sign of c, the sense of the velocity (14) is reversed*. 126. The problem of the integration of the equations of motion of a solid in the general case has engaged the attention of several mathematicians, but, as might be anticipated from the complexity of the question, the physical meaning of the results is not easily grasped 1. In what follows we shall in the first place inquire what simplifications occur in the formula for the kinetic energy, for special classes of soUds, and then proceed to investigate one or two particular problems of considerable interest which can be treated without difficult mathematics. The general expression for the kinetic energy contains, as we have seen, twenty-one coefficients, but by the choice of special directions for the co-ordinate axes, and a special origin, these can be reduced to fifteen J. * The substance of this Art. is taken from a paper, " On the Free Motion of a Solid through an Infinite Mass of Liquid," Proc. Lond. Math. 8oc. t. viii. (1877). Similar results were obtained independently by Craig, "The Motion of a Solid in a Fluid,*' Amer. Joum. of Math. t. ii. (1879). f For references see Wien, Lehrbuch d. Hydrodynamik, Leipzig, 1900, p. 164. X Cf. Clebsoh, "Ueber die Bewegung eines Korpers in einer Fltissigkeit," Math. Ann. t. iii. p. 238 ( 1 870). This paper deals with the * reciprocal ' form of the dynamical equations, obtained by substituting from Art 122 (4) in Art. 120 (1). 125-126] Hydrokinetic Symmetries 163 The most symmetrical way of writing the general expression is 2T = Au^ + Bv^ + Gw^ + 2A'vw ^2B'wu^ 20 'uv + Pp^ + Qq^ + Rr^ + 2P'qr + 2Q'rp + 2R'pq + 2Lup + 2Mvq + 22Vw + 2-F (w + w^j) + 2G {wp + wr) + 2H (uq + vp) + 2F' {vr - wq) + 2G' {wp - wr) + 2H' (uq - vp) (1) It has been seen that we may choose the directions of the axes so that A\ B\ C = 0, and it may easily be verified that by displacing the origin we can further make F\ G\ J?' = 0. We shall henceforward suppose these simpUfications to have been made. 1°. If the sohd has a plane of symmetry, it is evident from the con- figuration of the relative stream-lines that a translation normal to this plane must be one of the permanent translations of Art. 124. If we take this plane as that of xy, it is further evident that the energy of the motion must be unaltered if we reverse the signs of w, p, q. This requires that P\ Q\ L, M, N^ H should vanish. The three screws of Art. 125 are now pure rotations, but their axes do not in general intersect. 2"^. If the body has a second plane of symmetry, at right angles to the former one, we may take this as the plane xz. We find that in this case R' and G must also vanish, so that 2T = Au^ + Bv^ + Cw^ + Pp^ + Qq^ + Rr^ -\-2F(vr + wq). . .(2) The axis of x is the axis of one of the permanent rotations, and those of the other two intersect it at right angles, though not necessarily in the same point. 3°. If the body has a third plane of symmetry, say that of yz, at right angles to the two former ones, we have 2T = Au^ + Bv^ + Cw^ + Pp^ -{-Qq^ + Rr^ (3) 4°. Returning to (2°), we note that in the case of a solid of revolution about Oxy the expression for 2T must be unaltered when we write v, j, — w, — r for w, r, V, q, respectively, since this is equivalent to rotating the axes of y, z through a right angle. Hence 5 = (7, = i2, J = 0; and therefore 2T = 4w« + 5 (v« + w^) + Pp^^Q {q^ + r«) (4)* The same reduction obtains in some other cases, for example when the sohd is a right prism whose section is any regular polygon f. This is seen at once from the consideration that, the axis of x coinciding with the axis of the prism, it is impossible to assign any uniquely symmetrical directions to the axes of y and z. * For the solution of the equations of motion in this case see Greenhill, "The Motion of a Solid in Infinite Liquid under no Forces," Amer. J, of Math, t. xx. (1897). t See Larmor, "On Hydrokinetic Symmetry," Quart. Joum. Math. t. xx. (1885). 11—2 164 Motion of Solids throiigh a Liquid [chap, vi 5^. If, in the last case, the form of the solid be similarly related to each of the co-ordinate planes (for example a sphere, or a cube), the expression (3) takes the form 2r = 4 (w2 ^- v2 + w;2) + p (p2 + ya ^_ ^2) (5) This again may be extended, for a like reason, to other cases, for example any regular polyhedron. Such a body is practically for the present purpose 'isotropic,' and its motion will be exactly that of a sphere under similar conditions. 6°. We may next consider another class of cases. Let us suppose that the body has a sort of skew symmetry about a certain axis (say that of x), viz. that it is identical with itself turned through two right angles about this axis, but has not necessarily a plane of symmetry*. The expression for 2T must be unaltered when we change the signs of v, w, q, r, so that the coefficients Q\ R\ G, H must all vanish. We have then 2T « Au^ + Bv^ + Cw^ + Pp« + Qq^ + fir« + 2P'qr + 2Lup -h 2Mvq + 2Nw + 2J (vr + wq) (6) The axis of a; is one of the directions of permanent translation ; and is also the axis of one of the three screws of Art. 125, the pitch being — LJA, The axes of the two remaining screws intersect it at right angles, but not in general in the same point. 7°. If, further, the body be identical with itself turned through one right angle about the above axis, the expression (6) must be unaltered when V, J, — w;, — r are written for w, r, v, j, respectively. This requires that B = C, Q = fi, P' = 0, If = iV, J = 0. Hencet 2T ==Au^-\-B {v^ + w^) + Pp^-\-Q (g* + r^) + 2Lup + 2M (vq ■\-wr), . . (7) The form of this expression is unaltered when the axes of y, z are turned in their own plane through any angle. The body is therefore said to possess hehcoidal symmetry about the axis of x, 8^. If the body possess the same properties of skew symmetry about an axis intersecting the former one at right angles, we must evidently have 2T^A (tt« + t;« + !£;«) -f- P (p* + ?« + r«) + 2L (j9w + ?v + rw). . .(8) Any direction is now one of permanent translation, and any hne drawn through the origin is the axis of a screw of the kind considered in Art. 125, of pitch — LjA, The form of (8) is unaltered by any change in the directions of the axes of co-ordinates. The solid is therefore in this case said to be 'helicoidally isotropic' * A two-bladed screw-propeller of a ship is an example of a body of this kind. t This result admits of the same kind of generalization as (4), e,g. it applies to a body shaped like a screw-propeller with thrtt symmetrically- disposed blades. The integration of the equations of motion is discussed by Greenhill, "The Motion of a Solid in Infinite Liquid," Amer. J, of Maih. t. zxviii. p. 71 (1906). 126-127] Solid of Revolution 165 127. For the case of a solid of revolution, or of any other form to which the formula 2T^ 4w« + 5 (v* + w^) + Pp« + Q (?* + r2) (1) applies, the complete integration of the equations of motion was effected by Ejrchhoff * in terms of elliptic functions. The particular case where the solid moves without rotation about its axis, and with this axis always in one plane, admits of very simple treatment !» and the results are very interesting. If the fixed plane in question be that of xy we have p, q, w = 0, so that the equations of motion, Art. 124 (1), reduce to * * , (2) Let z, y be the co-ordinates of the moving origin relative to fixed axes in the plane (xy) in which the axis of the soUd moves, the axis of z coinciding with the line of the resultant impulse (J, say) of the motion ; and let 6 be the angle which the line Ox (fixed in the solid) makes with z. We have then Au = I cos 0, Bv = — I BtaOy r = 6, The first two of equations (2) merely express the fixity of the direction of the impulse in space ; the third gives g6f + ^^^^^/*sinflcose = (3) « We may suppose, without loss of generality, that A> B, If we write 2fl = ^, (3) becomes , ^+-^^^— 8m& = 0, (4) which is the equation of motion of the common pendulum. Hence the angular motion of the body is that of a ' quadrantal pendulum,' i,e, a body whose motion follows the same law in regard to a quadrant as the ordinary pendulum does in regard to a half-circumference. When has been determined from (3) and the initial conditions, z, y are to be found from the equations / J z = tt cos ^ — v sin fl = -1- cos* ^ + « sin* 6, y = w sin d + t; cosfl « f -J— ^ j sin fl cos d = y S, (5) * he ante p. 151. t See Thomson and Tait, Art. 322; Qreenhill, "On the Motion of a Cylinder through a Friotionless Liquid under no Forces,*' Mea9, of Math, t. ix. (1880). 166 Motion of Solids throtigh a Liquid [chap, vi the latter of which gives l-jO, (6) and is otherwise obvious, the additive constant being zero since the axis of z is taken to be coincident with, and not merely parallel to, the line of the impulse /. Let us first suppose that the body makes complete revolutions, in which case the first integral of (3) is of the form fi« = a>2 (1 - ** sin* e), (7) ^^^^^^ *'==4w-S (^) Hence, reckoning t from the position fl = 0, we have = f ' — T = F(k,0),.: (9) cot in the usual notation of elUptic integrals. If we eUminate t between (5) and (7), and then integrate with respect to 0, we find the origin of z being taken to correspond to the position 6 = 0. The path can then be traced, in any particular case, by means of Legendre's Tables. See the curve marked I on the opposite page. If, on the other hand, the solid does not make a complete revolution, but oscillates through an angle a on each side of the position ^ = 0, the proper form of the first integral of (3) is d^ = a>>(l-'^l^) (11) where sin* a = -j — -^ • ■?« (12) A — B P If we put sin ^ = sin a sin 0, 2 this gives -^^ = ^-g-- (1 — sin* a sin*0), whence -= — = F (sin a, 0) (13) sma ^ Transforming to as independent variable, in (5), and integrating, we find IT ^\ z = -^- sin a . -P (sin a, 0) f- cosec a . E (sin a, 0), | ^ ^ I. ...(14) y = -V cos ^p. 127] Solid of Revolution 167 The path of the point is now a sinuous curve crossing the line of the impulse at intervals of time equal to a half-period of the angular motion. This is illustrated by the curves III and IV of the figure. There remains a critical case between the two preceding, where the solid just makes a half-revolution, having as asymptotic limits the two values 4 168 Motion of Solids throtigh a Liquid [chap, vi ± Jtt. This case may be obtained by putting A; = 1 in (7), or a = Jtt in (11) ; and we find d' = ai 008 d, ; (15) ad = log tan (Jtt + ^6), (16) = D- log ^^ (i^ + i^) - ^ sin 0, ^ ^ y (17) z = y = ^ cos 0, See the curve II of the figure*. It is to be observed that the above investigation is not restricted to the case of a soHd of revolution; it applies equally well to a body with two perpendicular planes of symmetry, moving parallel to one of these planes, provided the origin be properly chosen. If the plane in question be that of xy, then on transferring the origin to the point {F/B, 0, 0) the last term in the formula (2) of Art. 126 disappears, and the equations of motion take the form (2) above. On the other hand, if the motion be parallel to zx we must transfer the origin to the point (— F/C, 0, 0). The results of this Article, with the accompanying diagram, serve to exemplify the statements made near the end of Art. 124. Thus the curve IV illustrates, with exaggerated amplitude, the case of a sUghtly disturbed stable steady motion parallel to an axis of permanent translation. The case of a shghtly disturbed unstable steady motion would be represented by a curve contiguous to II, on one side or the other, according to the nature of the disturbance. 128. The mere question of the stabiUty of the motion of a body parallel to an axis of symmetry may of course be more simply treated by approximate methods. Thus, in the case of a body with three planes of symmetry, as in Art. 126, 3°, sUghtly disturbed from a state of steady motion parallel to a;, we find, writing u = Uq-\- u\ and assuming u\ v, w, p, q, r to be all small. ^ ....(1) * In order to bring out the peculiar features of the motion, the curves have been drawn for the somewhat extreme case of A =5B. In the case of an infinitely thin disk, without inertia of its own, we should have A/B=co ; the curves would then have cusps where they meet the axis of y. It appears from (5) that ± has always the same sign, so that loops cannot occur in any case. In the various cases figured the body is projected always with the same impulse, but with different degrees of rotation. In the curve I, the maximum angular velocity is ^2 times what it is in the critical case 11 ; whilst the curves III and IV represent oscillations of amplitude 45^ < and IS*' respectively. 127-129] Solid of Revolution 169 Hence ^ A» "^ ^W — "•* » = 0, with a Bimilar equation for r, and ^i*w A{A-C) ^ - ,„, C-^ + — ^-g ^V«'=0. (2) with a similar equation for q. The motion is therefore stable only when A is the greatest of the three quantities A, B, C. It is evident from ordinary Dynamics that the stability of a body moving parallel to an axis of symmetry will be increased, or its instability (as the case may be) wUl be diminished, by communicating to it a rotation about this axis. This question has been examined by Greenhill*. Thus, in the case of a solid of revolution slightly disturbed from a state of motion in which u and p are constant and the remaining velocities are zero, if we neglect squares and products of small quantities the first and fourth of equations (1) of Art. 124 give du/cU=0, dp/dt=0, whence «=«0' P=Po* (3) say, where Uq, p^ are constants. The remaining equations then take, on substitution from Art. 126 (3), the forms J^r^-l>o«^j=-^V» B\^ + p^v\=Au^ (4) Q^HP'Q)P^=-(A-B)u^w, Q^-{P-Q)p^^{A-B)UoV. ....(6) If we assume that v, tr, g, r vary as e*'*, and eliminate their ratios, we find e<r«±(P-2G)l>o<r-{(P-G)V+^(^-5)V}=0. (6) The condition that the roots of this should be real lb that should be positive. This is always satisfied when A > B, and can be satisfied in any case by giving a sufficiently great value to p^. This example illustrates the steculiness of fiight which is given to an elongated projectile by rifling. 129. In the investigation of Art. 125 the term 'steady' was used to characterize modes of motion in which the 'instantaneous screw' preserved a constant relation to the moving sohd. In the case of a solid of revolution, however, we may conveniently use the term' in a somewhat wider sense, extending it to motions in which the vectors representing the velocities of translation and rotation are of constant magnitude, and make constant angles with the axis of symmetry and with each other, although their relation to points of the solid not on the axis may continually vary. * "Fiiiid Motion between Confocal Elliptic Cylinders, ^c." Quart. Joum. Math. t. zvi. (1879). 170 Motion of Solids throtigh a Liquid [chap, vi The conditions to be satisfied in this case are most easily obtained from the equations of motion of Art. 124, which become, on substitution from Art. 126 (4), A%=B(rv.^), Pf=0. \ B^^Bjnv-Am, Q^^^(A-B)uw- P-Q)pr,\ (1) B^^^Agu-Bpv, gJ= (A-B)uv + (P-Q)n'^ It appears that j} is in any case constant, and that g' +r* will also be constant provided v/q=tD/r, =*, say (2) This makes dufdt =0, and v" +t&' = const. It follows that k will also be constant; and it only remains to satisfy the equations kB^=(kBp-Au)r, Q^= -{{A -B)ku+{P-Q)p}r, These will be consistent provided kB {{A -B) ku + {P -Q) p) +Q {kBp -Au)=0, , u kBP .^. ^^'^'^^^ p=AQ-k^B(A-B) <^) Hence by variation of k we obtain an infinite number of possible modes of steady motion, of the kind above defined. In each of these the instantaneous axis of rotation and the direction of translation of the origin are in one plane with the axis of the solid. It is easily seen that the origin describes a helix about the line of the impulse. These results are due to Kirchhoff. 130. The only case of a body possessing helicoidal property, where simple results can be obtained, is that of the 'isotropic helicoid' defined by Art. 126 (8). Let be the centre of the body, and let us take as axes of co-ordinates at any instant a line Ox parallel to the axis of the impulse, a line Oy drawn outwards from this axis, and a line Oz perpendicular to the plane of the two former. If I and K denote the force- and couple-constituents of the impulse, we have Au + Lp= f = /, Av-i- Lq = 7) =0y Aw + Lr = ^ = 0, Pp -^ Lu = X = K, Pq + Lv = fjL = 0, Pr -{- Lw = v = Iw, where w denotes the distance -of from the axis of the impulse. Since AP — L^^Oy the second and fifth of these equations shew that v = 0, q = 0. Hence m is constant throughout the motion, and the remaining quantities are also constant; in particular PI - LK LIw = 1/ = JrnJ 129-132] Helicoidal Symmetry 171 The origin therefore describes a helix about the axis of the impulse, of pitch I L' This example is due to Kelvin*. a 131. Before leaving this part of the subject we remark that the preceding theory applies, with obvious modifications, to the acyclic motion of a liquid occupying a cavity in a moving solid. If the origin be taken at the centre of inertia of the liquid, the formula for the kinetic energy of the fluid motion is of the type 2T = m (w2 + v« + w^) + Pp* + Q?* + Rr« + 2V'qr + 2QVp + ^Kjq. . . (1) For the kinetic energy is equal to that of the whole fluid mass (m), supposed concentrated at its centre of inertia and moving with this point, together with the kinetic energy of the motion relative to the centre of inertia. The latter part of the energy is easily proved by the method of Arts. 118, 121 to be a homogeneous quadratic function of j9, q, r . Hence the fluid may be replaced by a solid of the same mass, having the same centre of inertia, provided the principal axes and moments of inertia be properly assigned. The values of the coefficients in (1), for the case of an ellipsoidal cavity, may be calcu- lated from Art. 110. Thus, if the axes of x, y, z coincide with the principal axes of the ellipsoid, we find p. ,.».,» '-55$1, ,»<$=^'. ^^^■. p-.*,E'=a Case of a Perforated Solid, 132. If the moving solid have one or more apertures or perforations, so that the space external to it is multiply-connected, the fluid may have a motion independent of that of the solid, viz. a cyclic motion in which the circulations in the several irreducible circuits which can be drawn through the apertures may have any given constant values. We will briefly indicate how the foregoing methods may be adapted to this case. * Lc. ante p. 158. It is there pointed out that a solid of the kind here in question may be constructed by attaching vanes to a sphere, at the middle points of twelve quadrantal arcs drawn BO as to divide the surface into octants. The vanes are to bo perpendicular to the surface, and are to be inclined at angles of 45° to the respective arcs. Larmor {l.c. ante p. 163) gives another example. **If . . .we take a regular tetrahedron (or other regular solid), and replace the edges by skew bevel faces placed in such wise that when looked at from any comer they all slope the same way, we have an example of an isotropic helicoid.** For some further investigations in the present connection sec a paper by Biiss Fawcett, "On the Motion of Solids in a Liquid," Quart. Jonm. Math. t. xxvi. (1893). 172 Motion of Solids through a Liquid [chap, vi Let #c, #c', #c", ... be the circulations in the various circuits, and let So*, So*', Sot", ... be elements of the corresponding barriers, drawn as in Art. 48. Further, let I, m, w denote the direction-cosines of the normal, drawn towards the fluid at any point of the surface of the sohd, or drawn on the positive side at any point of a barrier. The velocity-potential is then of the form where ^ = ^i + t;^« + «^s + ;%i + 8X2 + rXs.l qj The functions ^i> ^2> ^S) Xi> Xs> Xs ^^^ determined by the same conditions as in Art. 118. To determine co, we have the conditions : (1°) that it must satisfy V\o = at all points of the fluid ; (2°) that its derivatives must vanish at infinity ; (3°) that dw/dn must = at the surface of the solid ; and (4^) that CO must be a cyclic function, diminishing by unity whenever the point to which it refers completes a circuit cutting the first barrier once (only) in the positive direction, and recovering its original value whenever the point completes a circuit not cutting this barrier. It appears from Art. 52 that these conditions determine co save as to an additive constant. In like manner the remaining functions co', co", ... are determined. By the formula (5) of Art. 55, twice the kinetic energy of the fluid is equal to -pjj{<f>+<f>o)^{<f> + <f>o)d8 - pf^jj g^ (^ + ^0) *y - P*^' j^i^ +<f>o)da'- (2) Since the cycUc constants of <f> are zero, and since d<f>QJdn vanishes at the surface of the solid, we have, by Art. 54 (4), M^+'//l*'+"'//l!^+ •••-//* ysds-o. on Hence (2) reduces to -,jl^^iS-^jj%'i,-^'ij^i,'- (3) Substituting the values of ^, ^0 from (1) we find that the kinetic energy of the fluid is equal to T + Z, (4) where T is a homogeneous quadratic function of w, v, w, p, y, r, of the form defined by Art. 121 (2) (3), and 2K = (#c, k) k^ + (#c', #c') k'« + . . . + 2 {k, k')kk' + ..., . . .(6) V (6) 13^133] Perforated Solid 173 where, for example, {k, k)=-p fj^ da, The identity of the different forms of {k, k) follows from Art. 64 (4). Hence the total energy of fluid and solid is given by T = ® + ii:, (7) where 'ST is a homogeneous quadratic function of Uy v, w, p, q, r of the same form as Art. 121 (8), and K is defined by (5) and (6) above. 133. The 'impulse' of the motion now consists partly of impulsive forces applied to the sohd, and partly of impulsive pressures pK, pK, pK*\ . . . applied uniformly (as explained in Art. 54) over the several membranes which are supposed for a moment to occupy the positions of the barriers. Let us denote by ^i, 171, Jj, Aj, ftj, i/j the components of the extraneous Impulse applied to the solid. Expressing that the x-component of the momentum of the solid is equal to the similar component of the total impulse acting on it, we have = ii + P M(«^i + • • • + PXi+ "• + f<" + ...) S- dS where, as before, Tj denotes the kinetic energy of the solid, and T that part of the energy of the fluid which is independent of the cyclic motion. Again, considering the angular momentum of the solid about the axis of x, 1 = Ai - pjj((f> + ^0) (wy - mz) dS (5) 174 Motion of Solids through a Liquid [chap, vi Hence, since "ST = T + T^, we have *.-f-'«//"i'^-w/"'i^--- By virtue of Lord Kelvin's extension of Green's Theorem, already referred to, these may be written in the alternative forms Adding to these the terms due to the impulsive pressures applied to the barriers, we have, finally, for the components of the total impulse of the motion*, where, for example, ^•-^■//('+l')*'+^'//('+^')*''+-. It is evident that the constants lo> 'yo* ^o> ^, /^o> ^o are the components of the impulse of the cychc fluid motion which would remain if the solid were, by forces applied to it alone, brought to rest. By the argument of Art. 119, the total impulse is subject to the same laws as the momentum of a finite d3mamical system. Hence the equations of motion of the solid are obtained by substituting from (5) in the equations (1) of Art. 120t. 134. As a simple example we may take the case of an annular solid of revolution. If the axis of x coincide with that of the ring, we see by reasoning of the same kind as in Art. 126, 4° that if the situation of the origin on this axis be properly chosen we may write 2T = Au^-hB {v^ + w^) + Pp^-hQ {q^ + r«) + (k, k) kK . . .(1) Hence f , 77, ^ = .iw + fo, B% ^^l A, ^, 1/ = Pp, Qq,Qr (2) * Cf. Sir W. Thomson, Ic. ante p. 158. t This conclusion may be verified by direct calculation from the preesure-formula of Art. 20; see Bryan, " Hydrodynamical Proof of the Equations of Motion of a Perforated Solid, Phil Mag. May 1893. da' -h • ..(6) »» J 133-135] Components of Impvlse lib Substituting in the equations of Art. 120, we find dpldt = 0, or j? = const., as is otherwise obvious. Let us suppose that the ring is slightly disturbed from a state of motion in which v, w, p, q, r are zero, i.e. a steady motion parallel to the axis. In the beginning of the disturbed motion V, Wy p, q, r will be small quantities whose products we may neglect. The first of the equations referred to then gives du/dt = 0, or w = const., and the remaining equations become B^ = -(Au + Ur, Q^=-{iA-B)u + io}^^ B^= {Au + io)q, g|= {{A-B)u + Qv.\ Eliminating r, we find BQ^, = -iAu + ^o){{A-B)u + i^}v. ., (4) Exactly the same equation is satisfied by w. It is therefore necessary and sufficient for stability that the coefficient of v on the right-hand side of (4) should be negative ; and the time of a small oscillation, when this condition is satisfied, is* g^ r SQ -li (5) We may also notice another cane of steady motion of the ring, viz. where the impulse reduces to a couple about a diameter. It is easily seen that the equations of motion are satisfied by (, i;, (, X, /a =0, and v constant ; in which case u= -(o/A^ r= const. The ring then rot-ates about an axis in the plane yz parallel to that of z, at a distance u/r from itf. Equation of Motion in Generalized Co-ordinates, 135. When we have more than one moving soUd, or when the fluid is bounded, wholly or in part, by fixed walls, we may have recourse to Lagrange's method of * generahzed co-ordinates.' This was first apphed to hydrodynamical problems by Thomson and TaitJ. In any dynamical system whatever, if ^, rj, J are the Cartesian co-ordinates at time t of any particle m, and X, Y, Z the components of the total force acting on it, we have of course mf=Z, mrj=Y, ml=^Z (1) ♦ Sir W. Thomson, l.c, ante p. 168. t For further investigations on this subject we refer to papers by Basset, *'0n the Motion of a Ring in an Infinite Liquid," Proc. Camb, Phil. Soc. t. vi. (1887), and Miss Fawcott, Ix. ante p. 171. ; Natural Philowpky (Ist ed.), Oxford, 18A7, Art. 331. 176 Motion of Solids throv^h a ^Liquid [chap, vi Now let ^ + A^, ^i + Aiy, J + AJ be the co-ordinates of the same particle, at time f, in any arbitrary motion of the system differing infinitely little from the actual motion, and let us form the equation Sw (^Af + iyAiy + t'AO = S (ZA^ + YLri + ZAO, (2) where the summation I! embraces all the particles of the system. This follows at once from the equations (1), and includes these, on account of the arbitrary character of the variations Af , At;, A J. Its chief advantages, how- ever, consist in the extensive elimination of internal forces which, by imposing suitable restrictions on the values of Af , At;, A^, we are able to effect, and in the facilities which it affords for transformation of co-ordinates. It is to be noticed that SO that the symbols d and A are commutative. The systems ordinarily contemplated in Analytical Dynamics are of finite freedom; i.e. the position of every particle is completely determined when we know the values of a finite number of independent variables or 'generalized co-ordiiiates' j'l, j^a* • • • in^^^ that, for example, . 3^ . _^ a^ . _^ _^ a^ . ^^ai;^^"'a^'^-'----'arn*- ^ ^^ The kinetic energy can then be expressed as a homogeneous quadratic function of the 'generaUzed velocity-components' ^i, ft* • • • ?n> thus 2r = ii^gi' + ^229t* + . . . + 2iii,?ig2 + . . . , (4) where (5) The quantities ii^^, -4„ are called the 'inertia-coefficients' of the system; they are in general functions of the co-ordinates y^, y,* ••• ?n- Again, we have 2 (X^^ + Y^r| + ZAO = g^A?, + Q^^q^ + . . . + Qn^qn. • • • .(6) -k- 0,.2(x|+r| + z|) ,7) The quantities Q^ are called the 'generalized components of force.' In the case of a conservative system we have «— 1^, <») 135] Generalized Co-ordinates 177 Also, from (3) and (5), 2m (^A^ + ^At; + ^AO = (^iiffi + ^i%q% + . . . + A^nin) A?i 4- + Mnl?! + -4n2?2 + • • • + A^^q^) Ajn = a^,^«^ + a^/**-^---+3f/?- ••••(^) or Sm (^Af + aJAt; + ^A^ = ^i A?i + pj Aj, + . . . + ?„ A?„, (10) where P^^dd ^^^^ The quantities Pr are called the 'generalized components of momentum' of the system. When T is expressed as in (4) as a homogeneous quadratic function of g^, g^* • • • 9«> we have 22' = Ptqi + Ptqt + . . . + Pnqn (12) Since 2m (^A^ 4- rj^rj + gAO = | 2m (f A^ + ^Ar? + ^AO - AT, . . (13) the transformation of (2) to generalized co-ordinates is easily completed by substitution from (9) and (6). The variations A^^ of the velocities cancel; and, equating coefficients of the independent variations ^q^ of the co-ordinates, we obtain n equations of the type* ddT_dT_ From (12) and (14) we derive 2 ^ = ftjj + Pxqx + p^qt + p^q^ + . . . + /»«?« + Pnqn + o~ ?! + 5 — ?« + • • • + o" W« = -^ + Cl?l + 02?2 + • . . + ©«3n, whence -^ = Q^q^ -t- 0a92 + . . . + Q„9n, ; (15) or, in the case of a conservative system |(r+7) = 0. (16) which is the equation of energy. * This somxnary of Lagrange's proof ia introduced merely to facilitate reference to the ▼arions steps, in the hydrodynamical investigation of the next Art. A proof by direct transforma- tion of co-ordinates, not involving the method of 'variations/ has been given by Hamilton {PhxL Trans. 1835, p. 96), Jacobi, Bertrand, and Thomson and Tait; see also Whittaker, Anal^ical Dynamics, Cambridge, 1904, p. 33. L.H. 12 178 Motion of Solids throttgh a Liquid [chap, vi If we multiply (2) by &, and integrate between the limits t^ and t^ , we find, having regard to (13), ''{AT + 2 {X£ii + YArj -f ZAO} dt = / -(17) If we now introduce the additional condition that in the varied motion the initial and final positions shall be respectively the same for each particle as in the actual motion, the quantities A^, A77, A^ will vanish at both limits, and the equation reduces to f ' {AT -f 2 (ZA^ + TAt? + ZAO} * = 0, (18) or, for a conservative system*, (19) In words, if the actual motion of the system between any two configura- tions through which it passes be compared with any slightly varied motion, between the same configurations, which the system is (by the application of suitable forces) made to execute in the same time, the time-integral of the 'kinetic potential' | V — T is stationary. In terms of generalized co-ordinates, the equation (18) takes the form ' (AT + GiAji + GaAy, + . . . + Qn^qn) dt = (20) This embraces the whole dynamics of the system in a mathematically compact form. From it Lagrange's equations can immediately be deduced ; cf. Art. 139. /: 136. Proceeding now to the hydrodynamical problem, let ji, jj, . . . j„ be a system of generalized co-ordinates which serve to specify the configuration of the solids. We will suppose, for the present, that the motion of the fluid is entirely due to that of the soUds, and is therefore irrotational and acyclic. In this case the velocity-potential at any instant will be of the form <f> = qi<f>i + q2<f>2 + . . . + qn<f>n9 (1) where <f>i,<f>%^ ... are determined in a manner analogous to that of Art. 118. The formula for the kinetic energy of the fluid is then 2T = - P \w g^ ^'^ = -A-nji* -f A2292* -f . . . 4- 2A129392 + . . . , . (2) ♦ Sir W. R. Hamaton, "On a General Method in Dynamics," PhiL Trans, 1834, 1836. t The name was introduced by Helmholtz, "Die physikalische Bedeutung des Pnncips der kleinsten Wirkung," CreUe, t. c. p. 137 (1886) [Wiss, Abh. t. iii. p. 203]. Whittaker, Analytical Dynamics, p. 38, reverses the sign. 135-136] Application to Hydrodynamics 179 where A„=-p|)V,^J^'<iS. A„^ - p jf<f>r^^dS=^ - p jj<f>,^-^dS, ..(3) the integrations extending over the instantaneous positions of the bounding surfaces of the fluid. The identity of the two forms of A„ follows from Green's Theorem. The coefficients A^^, A„ will in general be functions of the co-ordinates 9i, 9'2> • • - 9n* If we add to (2) twice the kinetic energy, Ti, of the solids themselves, we get an expression of the same form, with altered coefficients, say 2T = ^nSi* -f ^22?2* + . . . + 2A^^q^q^ + (4) It remains to shew that, although our system is one of infinite freedom, the equations of motion of the solids can, under the circumstances pre- supposed, be obtained by substituting this value of T in the Lagrangian equations, Art, 135 (14). We are not at liberty to assume this without further examination, for the positions of the various particles of the fluid are not determined by the instantaneous values yi, ?a, . . . J« of the co-ordinates of the solids. For instance, if the solids, after performing various evolutions, return each to its original position, the individual particles of the fluid will in general be found to be finitely displaced*. Going back to the general formula (2) of Art. 135, let us suppose that in the varied motion, to which the symbol A refers, the solids undergo no change of size or shape, and that the fluid remains incompressible, and has, at the boundaries, the same displacement in the direction of the normal as the solids with which it is in contact. It is known that under these conditions the terms due to the internal reactions of the solids will disappear from the sum S (ZAf -f YArj + ZAO. The terms due to the mutual pressures of the fluid elements are equivalent to or |]y VM + "A, + nAJ) *S + Jj Jy (^ + ^ + ?^) i« JjA, where the former integral extends over the bounding surfaces, and I, m, n denote the direction-cosines of the normal, drawn towards the fluid. The volume-integral vanishes by the condition of incompressibility W'^'d^^ dz "^ (^^ * As a simple example, take the case of a circular disk which is made to move, without rotation, so that its centre describes a rectangle two of whose sides are normal to its plane; and examine the displacements of a particle initially in contact with the disk at its centre. 12—2 180 Motion of Solids through a Liquid [chap, vi The surface-integral vanishes at a fixed boundary, where and in the case of a moving solid it is cancelled by the terms due to the pressure exerted by the fluid on the solid. Hence the symbols Xj F, Z may be taken to refer only to the extraneous forces acting on the system, and we may write 2 {X^^ -f FAt? + ZAO = Qi Aft + Q, Aft + . . . + QMn, .... (6) where Qi, ^2* . • . Cn denote the generalized components of extraneous force. The varied motion of the fluid has still a high degree of generality. We will now further limit it by supposing that while the soUds are, by suitable forces applied to them, made to execute an arbitrary motion, the fluid is left to take its own course in consequence of this. The varied motion of the fluid may accordingly be taken to be irrotational, in which case the varied kinetic energy T + AT of the system will be the same function of the varied co-ordinates q^ -f Aj,., and the varied velocities q^ + Aj^ , that the actual energy T is of j^ and g,.. Again, considering the particles of the fluid alone, we shall have, on the same supposition, = p jj<f> (IA| + mArj + nAO dS, where use has again been made of the condition (5) of incompressibility. By the kinematical condition to be satisfied at the boundaries, we have lAi + mArf -f nAf = - -^^ Aft - ^ Aft - . . . - -J^ Aj„, and therefore 2m(|Af.fi7A^4-eA0 = ^p||<^(^^^Aft-f^*Aft-f...-h^^ = (Aiift + Aijft 4- . . . -f Aj„4„) Aft -f (Asift + Ajjft + . • + A^n^n) Aft -f . . . + (A„ift -f A„2ft + . . . + A„ngn) Ay„ 9T 3T 3T = a^,^?^ + g^,^?* + '--+a9/?"' (^) by (1), (2), (3) above. If we add the terms due to the solids, we find that the relation (9) of Art. 135 still holds; and the deduction of Lagrange's equations dtdi.'d^,-^' ^^' then proceeds exactly as before. 136-137] Application to Hydrodynamics 181 As in Art. 135, these equations lead to or, in the case of a conservative system, r + F = const. (9) 137. As a first application of the foregoing theory we may take an example given by Thomson and Tait*, where a sphere is supposed to move in a Uquid which is limited only by an infinite plane wall. Taking, for simpUcity, the case where the centre moves in a plane perpendicular to that of the wall, let us specify its position at time t by rectangular co-ordinates x, y in this plane, of which y denotes distance from the wall. We have 2T = Ax*^By\ (1) where A and B are functions of y only, it being plain that the term iy cannot occur, since the energy must remain unaltered when the sign of i; is reversed. The values of A, B can be written down from the results of Arts. 98, 99, viz. if m denote the mass of the sphere, and a its radius, we have A = m-\- l-npa* (l + A^). -B = »» + I'rpo* (l + I ^). . .(2) approximately, if y be great in comparison with a. The equations of motion give im-x. >^,-K|^,|^).y. ,3, where X, Y are the components of extraneous force, supposed to act on the sphere in a line through the centre. If there be no extraneous force, and if the sphere be projected in a direction normal to the wall, we have ^ = 0, and By^ = const (4) Since B diminishes as y increases, the sphere experiences an acceleration from the wall. Again, if the sphere be constrained to move in a line parallel to the wall, we have y = 0, and the necessary constraining force is ^=-if^ (5) * Ix, ante p. 175. I t i I t (2) 182 Motion of Solids through a Liquid [chap, vi Since dAjdy is negative, the sphere appears to be attracted by the wall. The reason of this is easily seen by reducing the problem to one of steady motion. The fluid velocity will evidently be greater, and the pressure therefore less, on the side of the sphere next the wall than on the further side; see Art. 23. The above investigation will also apply to the case of two spheres projected in an unlimited mass of fluid, in such a way that the plane y = is a plane of symmetry in all respects. 138. Let us next take the case of two spheres moving in the line of centres. The kinematical part of this problem has been treated in Art. 98. If we now denote by X, y the distances of the centres of the spheres A^ B from some fixed origin in the line joining them, we have 2T=Lx^-2Mxy +Ny^ (1) where the coefficients L, M, N are functions of y -x, or c, the distance between the centres. Hence the equations of motion are where X, Y are the forces acting on the spheres along the line of centres. If the radii a, b are both smaU compared with c, we have, by Art. 98 (16), keeping only the most important terms, Z=m+f7rpa", Jbr=2»rp— 3-, N=m'+^npb^, (3) approximately, where m, m' are the masses of the two spheres. Hence to this order of approximation dL ^ dM ^ a»6» dN ^ Tc=^^ -d^=-^''P-^' d^=^- If each sphere be constrained to move with constant velocity, the force which must be applied to ^ to maintain its motion is ^ dM dM .. _ a*6' .• ,.. ^=-a^yiy-^)-S^'»9=«^P-^i/'- (4) This tends towards B, and depends only on the velocity of B. The spheres therefore appear to repel one another ; and it is to be noticed that the apparent forces are not equal and opposite unless x= ±,y. Again, if each sphere make small periodic oscillations about a mean position, the period being the same for each, the mean values of the first terms in (2) will be zero, and the spheres therefore will appear to act on one another with forces equal to 6«-p -^ Wl (5) where \xy] denotes the mean value of xy. Ji &, y differ in phase by less than a quarter- period, this force is one of repulsion, if by more than a quarter-period it is one of attraction. 137-139] Cydic Motion 183 Next, let B perform small periodic oscillations, while A is held at rest. The mean force which must be applied to jii to prevent it from moving is J=4'^[y*] («) where [y*] denotes the mean square of the velocity of B, To the above order of approxi- mation dN/dc \b zero ; on reference to Art. 98 we find that the most important term in it is - I2ir pa^b^c'f so that the force exerted on il is attractive, and equal to «Tp-,f OT (7) This result comes under a general principle enunciated by Kelvin. If we have two bodies immersed in a fluid, one of which (A) performs small vibrations while the other {B) is held at rest, the fluid velocity at the surface of B will on the whole be greater on the side nearer A than on that which is more remote. Hence the average pressure on the former side will be less than that on the latter, so that B will experience on the whole an attraction towards A. As practical illustrations of this principle we may cite the apparent attraction of a delicately-suspended card by a vibrating tuning-fork, and other similar phenomena studied experimentaUy by Guthrie* and explained in the above manner by Kelvin f. Modification of Lagrange^s Equalions in the case of Cyclic Motion, 139. We return to the investigations of Art. 135, with the view of adapting them to the case where the fluid has cyclic irrotational motion through channels in the moving solids, or (it may be) in an enclosing vessel, independently of the motion due to the soUds themselves. Let us imagine barrier-surfaces to be drawn across the several apertures. In the case of channels in a containing vessel we shall suppose these ideal surfaces to be fixed in space, and in the case of channels in a moving solid we shall suppose them to be fixed relatively to the soUd. Let 'j(, ;^', x'\ . . . be the fluxes at time t across, and relative to, the several barriers; and let Xi x'» x"? • • • be the time-integrals of these fluxes, reckoned from some arbitrary epoch, these quantities determining (therefore) the volumes of fluid which have up to the time t crossed the respective barriers. It will appear that the analogy with a dynamical system of finite freedom is still conserved, provided the quantities x, x'. x". .-be reckoned as generaUzed co-ordinates of the system, in addition to those (ji, ^2* • • • ?n) which specify the positions of the moving solids. It is obvious already that the absolute values of Xj x'» x"> • • • ^^^ ^^^ enter into the expression for the kinetic energy, but only their rates of variation. ♦ "On Approach caused by Vibration," Proc, Boy. 8oc, t. xix. (1860) [PA*/. Mag. Nov. 1870]. t Beprint of Papers on Electrogtatics, Ac. Art. 741. For references to further investigations, both experimental and theoretioal, by Bjerknes and others, on the mutual infiaence of oscillating spheres in a fluid, see Hicks, "Report on Recent Researches in Hydrodynamics," Brii. Am. Rep. 1882, pp. 52. . . ; Love, BncycL d, math, Wies. U iv. (3). pp. Ill, 112. 184 Motion of Solids through a Liquid [chap, vt In the first place, we may shew that the motion of the fluid, in any given configuration of the solids, is completely determined by the instantaneous values of ^1, jjj • • • 9n> X» X^ ic'^ ^^^ ^' there were two modes of irrotational motion consistent with these values, then, in the motion which is the difference of these, the boundaries of the fluid would be at rest, and the flux across each barrier would be zero. The formula (5) of Art. 55 shews that under these conditions the kinetic energy must vanish. It follows that the velocity-potential can be expressed in the form Here <f>^ is the velocity-potential of a motion in which q^ alone varies and the flux across each barrier is accordingly zero. Again Q is the velocity- potential of a motion in which the solids are all at rest, whilst the flux through the first aperture is unity, and that through every other aperture is zero. It is to be observed that <j>x,<f>^, . . . ^n, O, Q!, . . . are in general all of them cyclic functions, which may however be treated as single-valued, on the conventions of Art. 50. The kinetic energy of the fluid is given by the expression »-///i!i)'-(rMi)]*'*^' '^' where the integral is taken over the region occupied by the fluid at the instant under consideration. If we substitute from (1) we obtain T as a homogeneous quadratic function of g^, g'j, . . . j^, ;^, y^^ ;)^", . . . with coefficients which depend on the instantaneous configuration of the solids, and are there- fore functions of ji, y^, . . . ?„ oiily. Moreover, we find, by Art. 53 (1), aT fffidAdCl d6dCl ^ d(f>dQ) , , , dn '"' where k, k\ ... are the cyclic constants of <f), and the first surface-integral is to be taken over the surfaces of the solids, and the remaining ones over the several barriers. By the conditions which determine Q, this gives the first equation of the system : 3^ = ^'^' a^'""^"' ^ ^ These shew that p/c, p/c', . . . are to be regarded as the generalized components of momentum corresponding to the velocity-components x» x'» • • • » r^p^- tively. We have recourse to the general Hamiltonian formula* (17) of Art. 135. * It is possible to arrange an investigation on the Lagrangian plan, parallel to that of Art. 136, but the proof of the formulae corresponding to (5) below involves some rather delicate considerations. 139] Application of Hamiltonian Method 185 We will suppose that the varied motion of the solids is subject only to the condition that the initial and final configurations are to be the same as in the actual motion ; also that the initial position of each particle of the fluid is the same in the two motions. The expression wiU accordingly vanish at time ^q, but not in genera] at time t-i, in the absence of further restrictions. We will now suppose that the varied motion of the fluid is irrotational, and accordingly determined by the instantaneous values of the varied generalized co-ordinates and velocities. Considering the particles of the fluid alone, we have = p \\<f> (iAf + mL-q + nAO dS + pK jj{lAi + mAiy + nAO dcr + PkJJ{IM + m^ri + nAO (fo' +...,. .(4) where I, m, n are the direction-cosines of the normal to an element of the bounding surface, drawn towards the fluid, or (as the case may be) of the normal to an element of a barrier, drawn in the direction in which the corresponding circulation is estimated. At time ti we shall have lAi + wAt; + nAC = at the surface of the solids, as well as at the fixed boundaries. Again, if AB represent one of the barriers in its position at time ti, and if A'B' represent the locus at the same instant, in the varied motion, of those particles which in the actual motion occupy the position AB, the volume included between AB and A'B' will be equal to the corresponding Ax, whence jj(lAi + mAri + nAO da -= Ax, jj(lM + mArj + nAO At' = Ax', (5) The varied circulations are, from instant to instant, still at our disposal. We may suppose them to be so adjusted as to make A^, A^', . . . vanish at time tj . The right-hand member of Art. 135 (17) will accordingly vanish, and if we further suppose that the extraneous forces do on the whole no 186 Motion of Solids through a Liquid [chap, vi work when the boundary of the fluid is at rest, whatever relative displace- ments be given to the parts of the fluid, the formula reduces to [''{AT + QiAffi + Q^^q^ + . . . + QnAjn} * - (6) From this Lagrange's equations follow by a known process. We have Hence, by a partial integration, and remembering that by bypothesis A^i, Ajt, . . . iiq„, Ax, Ax', . . . vanish at the limits t^, ti, we find , fddT dT „ \ . , ddT . d dT . , ) ,, ^ (8) Since the values of Ajj, ^q^, ... A^n* ^X' ^x'» ••• within the range of integration are still arbitrary, their coefficients must separately vanish. We thus obtain n equations of the type dtdqr dqr~^" ^ ' togetherwith |gf = 0' llf' = ^' (^^^ 140. Equations of the types (9) and (10) present themselves in various problems of ordinary Dynamics, e,g, in questions relating to gyrostats, where the co-ordinates x> x'» • • • > whose absolute values do not afiect the kinetic or the potential energy of the system, are the angular co-ordinates of the gyrostats relative to their frames. The general theory of such systems has been treated by Routh*, Thomson and Taitf, and other writers. * On the Stability of a Given State of Motion (Adams Prizo Essay), London, 1877 ; Advanced Rigid Dynamics, 6th ed., London, 1905. t Natural Philosophy, 2nd ed.. Art. 319 (1879). See also Helmholtz, "Principien der Statik monooyclischer Systeme," CreUe, t, xcyii. (1884) [Wiss. Ahh, t. iiL p. 179]; Larmor, "On the Direct Application of the Principle of Least Action to the Dynamics of Solid and fluid Systems," Proc. Lond. Math. Sog. t. xv. (1884); Lamb, Art. "Dynamics, Analytical," Encyc, Brit. 10th ed. t. xzvii. p. 566 (1902), 11th ed. t. viii. p. 759 (1910); Whittaker, Analytical Dynamics, c. iii 139-141] Ignoratimi of Co-ordinates 187 We have seen that ^-r = p#c, x-r, = pk\ .... (11) and the integration of (10) shews that the quantities #c, k\ . . . are constants with regard to the time, as is otherwise known (Art. 50). Let ns write R=T^Pkx-pk'x'- (12) The equations (11), when written in full, determine ;^, ;^', . . . as linear functions of /c, K,... and qi,4t» • • • ?n 5 and by substitution in (12) we can express R as a homogeneous quadratic fimction of the same quantities, with coefficients which of course in general involve the co-ordinates Ji, ?2) • • • ?n* On this supposition we have, performing the arbitrary variation A on both sides of (12), and omitting terms which cancel by (11), ^^ A . . , 3^ A . , 3^ A , = g-r- A^i + . . . + g— Aji -h . . . — px^ — • • • J • • -(13) where, for brevity, only one term of each kind is exhibited. Hence we obtain 2n equations of the types dR^dT dR^dT dqr a?r' 3?r 3?r ' together with —=^px, diP^" ^^'' ^^^^ Hence the equations (9) may be written ddR dR ^ .^ ^. i^d^rwr^^' ^ ^ where the velocities x* x'> • • • corresponding to the * ignored' co-ordinates Xj x'> • • • have now been eliminated*. 141. In order to shew more explicitly the nature of the modification introduced by the cyclic motions into the dynamical equations, we proceed as follows. If we substitute in (12) from (15), we obtain r-«-('S^4*+-) ™ Now, remembering the composition of R, we may write for a moment R = -Ba,o + -Bi.i + ^,2> (18) where £2,0 is a homogeneous quadratic function of g^, 92? • • • 9n> -^.2 1^ & * This inyestigation is due to Routh, I.e. ; cf. Whittaker, Analytical Dynamics, Ait. 38. 188 Motimi of Solids through a Liquid [chap, vi homogeneous quadratic function of k,k\ . . . , and Bi,i is bilinear in these two sets of variables. Hence (17) takes the form T = B,.o~Bo.«, (19) or, as we shall henceforth write it, T = ^-|-ir, (20) where ^ and K are homogeneous quadratic functions of g^, gj* • • • ?n> ^^'^ of K^K'y . . . , respectively. It follows also from (18) that * = ®-^-A?i-i3292-----i3«?«, (21) where j8, , jSj, ... are linear functions of /c, #c', . . . , say ft = a^K + a^K* + . . . / ft = aj/c + OjV + . . . , ft = a^K + a„V + . . . . (22) The meaning of the coefficients a (in the hydrodynamical problem) appears from (15) and (21). We find dK PX ^ 3;7 "^ ^1 ?> "^ ^« 9a + . . . +a„'g„, •/ / • /J. / • V (23) which shew that a, is the contribution to the flux of matter across the first barrier due to unit rate of variation of the co-ordinate g,., and so on. If we now substitute from (21) in the equations (16) we obtain the general equations of motion of a 'gyrostatic system,' in the form* diWi'Wx +(1.2)^. + (1.3)j,+ ... + (l,n)g„ + g^ dK dm m ^,^ ,.. + (2,3)?,+ ... +(2,n)g, + g- = Q„ dm m , , ,.., , ox . i / ox . , *a^~a?;"^^"' ^^'"^^"' ^'*"^^"' ^^•^••' where (,, ,) = f. _ p. dqr dq. ^dq„ ^"' (24) (25) It is important to notice that (r, s) = — («, r), and (r, r) = 0. * These equations were first given in a paper by Sir W. Thomson, "On the Motion of Bigid Solids in a Liquid circulating irrotationaliy through perforations in them or in a Fixed Solid," Phil Mag. May 1873 [Ptipers, t. iy. p. 101]. See also 0. Neumann, Hydrodj/namische UnUf' wchungen (1883). 141-142] Equations of Motion 189 If in the equations of motion of a fuUy-specilBled system of finite freedom (Art. 135 (14)) we reverse the sign of the time-element 8<, the equations are unaltered. The motion is therefore reversible ; that is to say, if as the system is passing through any assigned configuration the velocities 9i, 92, . . . 9n ^^ all reversed, it will (if the forces be always the same in the same configuration) retrace its former path. It is important to observe that this statement does not in general hold of a gyrostatic system ; thus, the terms in (24) which are linear in j^, jg? • • • ^n change sign with 8^, whilst the others do not. Hence, in the present application, the motion of the solids is not reversible, unless indeed we imagine the circulations /c, k,,,, to be reversed simultaneously with the velocities g,, g,, ... g„*. If we multiply the equations (24) hj qj, q^, . . . qn i^ order, and add, we find, by an obvious adaptation of the method of Art. 135, I (® + ^) = Q,q, + Q^q^ + . . . + (?„?„, (26) or, if the system be conservative, -21; + ^+ 7 = const (27) 142. The results of Art. 141 may be applied to find the conditions of equilibrium of a system of solids surrounded by a Uquid in cyclic motion. This problem of * Kineto-Statics,' as it may be termed, is however more naturally treated by a simpler process. The value of <f) under the present circumstances can be expressed in the alternative forms ^ = x" + x'ii'+---> (1) <f> = KO) + #c'a>' + . . . ; (2) and the kinetic energy can accordingly be obtained as a homogeneous quad- ratic function either of %, %', . . . , or of k,k, . . . , with coefficients which are in each case fimctions of the co-ordinates Ji, g'2> • • • ?n which specify the configuration of the solids. These two expressions for the energy may be distinguished by the symbols Tq and K, respectively. Again, by Art. 55 (5) we have a third formula 2T = Pkx + PkY+ (3) The investigation at the beginning of Art. 139, shortened by the omission of the terms involving q^, q^, ... g„, shews that />'^= a-' p^ -sY' ^ ^ * Just as the motion of the axis of a top cannot be reversed unless we reverse the spin. 190 Motion of Solids through a Liquid [chap, vi Again, the explicit formula for K is ■ = (/c, k) k^ + (k\ /c') '<^'" + . . • + 2 (/c, /c') #<r#c' + . . • , ... .(5) where (k, k) = - p\^f^ At, (k, k') = - p (J ^ At = -p\\^ ^' , ■ ■ ■ -(6) and so on. Hence dK dK (>^, k)k+ (k, k') k' + 1 * • ^~ We thus obtain . dK P^=dK' PX = dK dK" (7) Again, writing Tq + K for 2T in (3), and performing a variation A on both sides of the resulting identity, we find, on omitting terms which cancel in virtue of (4) and (7)*, dqr^Wr"^ ^®^ This completes the requisite analytical formulae f. If we now imagine the solids to be guided from rest in the configuration {9j> 9t9 • • • 9n) to rest in an adjacent configuration the work required is QiAqi + OzAjg + . . . + Qn^9n9 where Qi,Q^, ... Qn are the components of extraneous force which have to be applied to neutrahze the pressures of the fluid on the solids. This must be equal to the increment AK of the kinetic energy, calculated on the supposition that the circulations k, k\.,. are constant. Hence e, = ^ (9) The forces representing the pressures of the fluid on the solids (when these are held at rest) are obtained by reversing the signs, viz. they are given by e'' = -i' (10) the solids therefore tend to move so that the kinetic energy of the cyclic motion diminishes. * It would be sufficient to assume either (4) or (7) ; the process then leads to an independent proof of the other set of formulae. f It may be noted that the function R of Art. 140 now reduces to - ^. 142-144] Kimto-Statics 191 In virtue of (8) we have, also, «''=!; (11) 143. A simple application of the equations (24) of Art. 141 is to the case of a sphere moving in a liquid which circulates irrotationally in a cyclic space with fixed boimdaries. If the radius a, say, of the sphere be small compared with its least distance from the fixed boundary, the formula (20) of Art. 141 becomes 2T=m (£^ +^* +«*) +K, (1) where x, y, z are the co-ordinates of the centre, and m denotes the mass of the sphere to- gether with half that of the fluid displaced by it ; sec Art. 92. To find K, the energy of the cyclic motion when the sphere is held at rest in its actual position, we note that if we equate x, y, i to the components u, v, Wy respectively, of the fluid velocity which would obtain at the point (x, |/, z) if the sphere were absent, and at the same time put m=2n-pa^ the resulting energy will be practically the same as that of the fluid when filling the region, whence 2irpa' (u^ +f^+v^) +^ =const., or K =const. -W (2) where W =^2irpa^ (w* +v* +tt'«) (3) Again the coefficients a^, a^, a^ of Art. 141 (22) denote the fluxes across the first barrier, when the sphere moves with unit velocity parallel to ar, y, z respectively. If we denote by O the flux across this barrier due to a unit simple-source at (x, y, z), then remembering the equivalence of a moving sphere to a double-source (Art. 92), we have ax,a„a3=K^> W ^^ i«» ^ (4) SO that the quantities denoted by (2, 3), (3, 1), (1, 2) in Art. 141 (24) vanish identically. The equations therefore reduce in the present case to mx=X + ^, iiiy=F + -g^, mz = Z-h-g^ (6) where X, F, Z are the components of extraneous force applied to the sphere. When X, Y, Z=0, the sphere tends to move towards places where the undisturbed velocity of the fluid is greatest. For example, in the case of cyclic motion round a fixed circular cylinder (Arts. 27, 64), the fluid velocity varies inversely as the distance from the axis. The sphere will therefore move as if under the action of a force towards this axis varying inversely as the cube of the distance. The projection of its path on a plane perpendicular to the axis will therefore be a Cotes' spiral*. 144. We may also notice one or two problems of Kineto-Statics, in illustration of the theory of Art. 142. * Cf. Sir W. Thomson, Ic. anU p. 188. 192 Motion of Solids through a Liquid [chap, vi It will be shewn in Art. 16? that the energy K of the cyclic fluid motion is propor- tional to the energy of a system of electric current-sheets coincident with the fixed boundaries, the current-lines being orthogonal to the stream-lines of the fluid. The electi'omagnetic forces between conductors carrying these currents are proportional * to the expressions on the right-hand of Art. 142 (10) with the signs reversed. Hence in the hydrodynamical problem the forces on the solids are opposite to those which obtain in the electrical analogue. In the particular case where the fixed solids reduce to infinitely thin cores, round which the fluid circulates, the current-sheets in question are practically equivalent to a system of electric currents flowing in the cores, regarded as wires, with strengths k, k, ... respectively. For example, two thin circular rings, having a common axis, will repel or attract one another according as the fluid circulates in the same or in opposite directions through themf. This might have been foreseen of course from the principle of Art. 23. Another interesting case is that of a number of open tubes, so narrow as not sensibly to impede the motion of the fluid outside them. If streams be established through the tubes, then as regards the external space the extremities will act as sources and sinks. The energy due to any distribution of positive or negative sources my^m^, ... is given, so far as it depends on the relative configuration of these, by the integral -\p\\<t>^da, (6) taken over a system of small closed surfaces surrounding m^, m,, ... respectively. If <f>i, <t>t, ... be the velocity-potentials due to the several sources, the part of this expression which is due to the simultaneous presence of 914, m, is -*''//(^l*-*.^')'^' (') which is by Green's Theorem equal to -f>jj<t>^ '-^^^- (8) Since the surface-integral of d<f>Jdn is zero over each of the closed surfaces except the one surrounding m^, we may ultimately confine the integration to this, and so obtain "''**// "^^^>=^«** (^) Since the value of <^| at m, is mi/^irfi^, where r,, denotes the distance between m^ and m,, we obtain, for the part of the kinetic energy which varies with the relative positions of the sources, the expression h'T: ^''^ The quantities m^, m^, ... are in the present problem equal to the fluxes xo» Xo^ • • • across the sections of the respective tubes, so that (10) corresponds to the form Tg of the * Maxwell, Electricity and Magnetism, Art. 573. t Tho theorem of this paragraph was given by Kirchhoff, l.c. ante p. 62. See also Sir W. Thomson, "On the Forces experienced by Solids immersed in a Moving Liquid," Proc. R, S. Edin. 1870 [Reprint, Art. xli.]; Boltzmami, **Ueber die Druckkr&fte welche auf Ringe wirksam Bind die in bewegte Flussigkeit tauchen," CreUe, t. Ixxiii (1871) [Wise. Abh, t. L p. 200.] 144] Kineto-StaticB 193 kinetio energy. The force apparently exerted by m^ on m,, tending to increase fn, is therefore, by Art. 142 (11), Hence two sources of like sign attract, and two of unlike sign repel, with forces varying inversely as the square of the distance*. This result, again, is easily seen to be in accord- ance with general principles. It also follows independently from the electric analogy, the tubes corresponding to Ampdre's * solenoids.' We here take leave of this branch of our subject. To avoid, as far as may be, the suspicion of vagueness which sometimes attaches to the use of 'generalized co-ordinates/ an attempt has been made in this Chapter to put the question on as definite a basis as possible, even at the expense of some degree of prolixity in the methods. To some writers f the matter has presented itself as a much simpler one. The problems are brought at one stroke imder the sway of the ordinary formulae of Dynamics by the imagined introduction of an infinite number of 'ignored co-ordinates,' which would specify the configuration of the various particles of the fluid. The corresponding components of momentum are assumed all to vanish, with the exception (in the case of a cyclic region) of those which are represented by the circulations through the several apertures. Prom a physical point of view it is difficult to refuse assent to such a generalization, especially when it has formed the starting-point of all the development of this part of the subject; but it is at least legitimate, and from the hydrodynamical standpoint even desirable, that it should be verified a posteriori by independent, if more pedestrian, methods. Whichever procedure be accepted, the result is that the systems con- templated in this Chapter are found to comport themselves (so far as the 'palpable' co-ordinates ft, ft* ••• ?n *r® concerned) exactly Uke ordinary systems of finite freedom. The further development of the general theory belongs to Analytical Dynamics, and must accordingly be sought for in books and memoirs devoted to that subject. It may be worth while, however, to remark that the hydrodynamical systems afEord extremely interesting and beautiful illustrations of the Principle of Least Action, the Reciprocal Theorems of Helmholtz, and other general dynamical theories. * Sir W. Thomaon, Ic t See Thomson and Tait, and Larmor, U. ciL ante p. 186. L. H. 13 CHAPTER VII VORTEX MOTION 145. Our investigations have thus far been confined for the most part to the case of irrotational motion. We now proceed to the study of rotational or 'vortex' motion. This subject was first investigated by Hehnholtz*; other and simpler proofs of some of his theorems were after- wards given by Kelvin in the paper on vortex motion abeady cited in Chapter in. We shall, throughout this Chapter, use the symbols f, ly, ^ to denote, as in Chapter in., the components of vorticity, viz. ^~a^"a^' '^"di'^di' ^"a5""a^ ^^ A line drawn from point to point so that its direction is everywhere that of the instantaneous axis of rotation of the fluid is called a 'vortex-line.' The differential equations of the system of vortex-lines are dx dv dz ,^, -y = ^ = — (2) ? V C If through every point of a small closed curve we draw the corresponding vortex-line, we mark out a tube, which we call a * vortex-tube.' The fluid contained within such a tube constitutes what is called a 'vortex-filament,' or simply a 'vortex.' Let ABC, A'B'C be any two circuits drawn on the surface of a vortex- tube and embracing it, and let AA' be a connecting line also drawn on the surface. Let us apply the theorem of Art. 32 to the circuit ABCAA'C'B'A'A and the part of the surface of the tube bounded by it. Since at every point of this surface, the Une-integral / {udx + vdy + wdz)y * "Ueber Integrale der hydrodynamischen Gleiohungen welche den Wirbelbewegungen enteprechen," CreOe, t. Iv. (1858) [Wiss, Ahh. t. i. p. 101]. 145] VorteooFUaments 195 taken round the circuit, must vanish ; ».e. in the notation of Art. 31 I (ABC A) + I {AA') + / {A'C'B'A') + I (A' A) = 0, which reduces to / (ABCA) = I {A'B'C'A'), Hence the circulation is the same in all circuits embracing the same vortex- tube. Again, it appears from Art. 31 that the circulation round the boundary of any cross-section of the tube, made normal to its length, is axr, where <«>> = (^ + ^* + i*) > is the resultant vorticity of the fluid, and a the infinitely small area of the section. Combining these results we see that the product of the vorticity into the cross-section is the same at all points of a vortex. This product is conveniently taken as a measure of the 'strength' of the vortex*. The foregoing proof is due to Kelvin ; the theorem itself was first given by Helmholtz, as a deduction from the relation i+l+i-» p' which follows at once from the values of ^, ly, t, given by (1). In fact writing, in Art. 42 (1), |^, iq, t, iot J7, F, Tf , respectively, we find lim^-mri-VnDdS^O, (4) where the integration extends over any closed surface lying wholly in the fluid. Applying this to the closed surface formed by two cross-sections of a vortex-tube and the part of the walls intercepted between them, we find cu^oT] =co2<72, where cu^, 0)2 denote the vorticities at the sections or^, c72, respectively. Kelvin's proof shews that the theorem is true even when ^, ly, t, are discontinuous (in which case there may be an abrupt bend at some point of a vortex), provided only that w, t), w are continuous. An important consequence of the above theorem is that a vortex-line cannot begin or end at any point in the interior of the fluid. Any vortex- lines which exist must either form closed curves, or else traverse the fluid, beginning and ending on its boundaries. Compare Art. 36. The theorem of Art. 32 (3) may now be enunciated as follows : The ^ circulation in any circuit is equal to the sum of the strengths of all the vortices which it embraces. * The circvlaiion round a vortex being the most natural measure of its intensity. la— 2 </./. 196 Vortex Motion [ch. vn 146. It was proved in Art. 33 that in a perfect fluid whose' density is either uniform or a function of the pressure only, and which is subject to forces having a single-valued potential, the circulation in any circuit moving with the fluid is constant. Applying this theorem to a circuit embracing a vortex-tube we find that the strength of any vortex is constant. If we take at any instant a surface composed wholly of vortex-lines^ the circulation in any circuit drawn on it is zero, by Art. 32, for we have 1$ -{- mrj + n^ = 8kt every point of the surface. The preceding Art. shews that if the surface be now supposed to move with the fluid, the circulation will always be zero in any circuit drawn on it, and therefore the surface will always consist of vortex-lines. Again, considering two such surfaces, it is plain that their intersection must always be a vortex-Une, whence we derive the theorem that the vortex-Unes move with the fluid. This remarkable theorem was first given by Helmholtz for the case of incompressibility ; the preceding proof, by Kelvin, shews that it holds for all fluids subject to the conditions above stated. The theorem that the circulation in any circuit moving with the fluid ia ^ -^ invariable constitutes t he sole and sufficient ft ppeal to Dynami cs which it is necessary to make in the investigations of this Chapter^ It is based on the hypothesis of a continuous distribution of pressure, and (conversely) implies this. For if in any problem we have discovered functions u, v, w of X, y, z, t, which satisfy the kinematical conditions, then, if this solution is to be also dynamically possible, the relation of the pressures about two moving particles A^ B must be given by the formula (2) of Art. 33, viz. B D fB j??+Q^jj2 =-gj {udx-\-vdy + wdz) (1) It is therefore necessary and sufficient that the expression on the right hand should be the same for all paths of integration (moving with the fluid) which can be drawn from A to B. This is secured if, and only if, the assumed values of u, v, w make the vortex-lines move with the fluid, and also make the strength of every vortex constant with respect to the time. It is easily seen that the argument is in no way impaired if the assumed values of u, v, w make ^, 77, ^ discontinuous at certain surfaces, provided only that w, v, w are themselves everywhere continuous. On account of their historical interest, one or two independent proofs of the preceding^ theorems may be briefly indicated, and their mutual relations pointed out. Of these proofe, perhaps the most conclusive is based upon a slight generalization of some equations given originally by Cauchy in the introduction to his great memoir on Waves*, and employed by him to demonstrate Lagrange's velocity-potential theorem. * 2.C. ante p. 10. 146] Persistence of Vortices 1»7 The equatioiia (2) of Art. 16 yield, on elimination of the function x ^7 crose-differentia- tion* 8u^ du^ 'bvhy dvh/ dwdz dw dz ^ dw^ dv^ dbde' dc db 569c dcdb db dc^ ^ db~ ~^ ~ ^ (where u, v, to have been written in place of dx/dt, dy/dt, dz/dt, respectively), with two symmetrical equations. .If in these equations we replace the differential coefficients of Uf V, to with respect to a, &, c, by their values in terms of differential coefficients of the same quantities with respect to x, y, z, we obtain ^d{y,z) .d{z,x) , .9(a?,y)_> \ ^a(6,c)'*"*'a(6,c)'*"^a(fe, c)"^«' ^d(a, b) ^^d{a, 6) "*"' 8(a, b) ""'*'• ; If we multiply these by dx/da, dz/db, dx/dCy in order, and add, then, taking account of the Lagrangian equation of continuity (Art. 14 (1)) we deduce the first of the following three symmetrical equations: * p p^da po 96 podc' ^=:^^+5?^+(5?y, y (3) p Po^ Po^ Po^' ' p Po ^ Po ^ Po ^ In the particular case of an incompressible fluid (p =po) these differ only in the use of the notation f , rj, ( from the equations given by Cauchy. They shew at once that if the initial values fo' '7o> Co ^^ ^^® component vorticities vanish for any particle of the fluid, then (, 17, { are always zero for that particle. This constitutes in fact Cauchy's proof of Lagrange's theorem. To interpret (3) in the general case, let us take at time ^ =0 a linear element coincident with a vortex-line, say Po Po Po where c is infinitesimal If we suppose this element to move with the fluid, the equations (3) shew that its projections on the co-ordinate axes at any other time will be given by asr, «y, &=€^, f^. €^, P P P ue, the element will still form part of a yortex-line, and its length (df, say) will vary as a>/p, where » is the resultant vortioity. But if a- be the cross-section of a vortex-filament having ds as axis, the product pads is constant with regard to the time. Hence the strength wr of the vortex is constant*. * See NaoBon, Mega, of Maih. t. iiL p. 120 (1874); Kirchhoff, Meckanik, c. xv. (1876); Stokes, Papers, t. ii p. 47 (1883). 198 Vortex Motion [oh. vn The proof given originally by Helmholtz depends on a system of three equations which, when generalized so as to apply to any fluid in which p is a function of p only, become* Dt\p) p dx pdy p^' Dt \pj pdx pdy p^' R(i\ = i^ 1^ ^^ Dt \pj "pdx pdy p dz' J (4) These may be obtained as follows. The dynamical equations of Art. 6 may be written, when a force-potential O exists, in the forms provided du ^ 8y' \ dw ^ dv' (6) (6) where 9* =u* +t)* +(«*. From the eeoond and third of these we obtain, «lit»ina.t ing ^' by oroea-diSerentiation, 3* d( 31 /a> d{\ du ,du ,/dv dtv\ Remembering the relation dx 4-^ + ^f- ^ dz =0, and the equation of continuity Dp Dt ^p{ fdu idx dv dy"- dm dz (7) we easily deduce the first of equations (4). To interpret these equations we take, at time t, a linear element whose projections on the co-ordinate axes are ^ n C P P P (9) where c is infinitesimaL If this element be supposed to move with the fluid, the rate at which bx is increasing is equal to the difference of the values of « at the two ends, whence Dbx __ ^ du Tj du ( du Dt p dx p dy p d^' It follows, by (4), that 5(*^-'f)=«' s(*»'-'j)=«-' s(»*-'^)=« <i«> * Nanson, l.c. 146] HdmhoUz' Equatimis 199 Helmholtz oonoludee that if the relations (9) hold at time t^ they will hold at time t-\-btt and so on, continually. The inference is, however, not quite rigorous; it is in fact open to the criticisms which Stokes* directed against various defective proofs of Lagrange's velocity-potential theorem f. By way of establishing a connection with Kelvin's investigation we may notice that the equations (2) express that the circulation is constant in each of three infinitety small circuits initially perpendicular, respectively, to the three co-ordinate axes. Taking, for example, the circuit which initially bounded the rectangle d6 dc, and denoting by A, B, C the areas of its projections at time t on the co-ordinate planes, we have 9 (6, C) d{b,c) (6, c) SO that the first of the equations referred to is equivalent :( to ^A +ffB + {C =&a6dc (11) As an application of the equations (4) we may consider the motion of a liquid of uniform vorticity contained in a fixed ellipsoidal vessel§. The formulae u—qz-ry, v^^rx-pZy w=py-qx (12) obviously represent a uniform rotation of the fluid as a solid within a spherical boundary. Transforming the co-ordinates and the corresponding velocities by homogeneous strain we obtain the formulae u _qz ry v _rx pz w _py gx ,,ox « ~ ~r jT* t "~ ~z zr » i ~~ t. it* ••••••••••••• •k*"^ acooaccoa as representing a certain motion within a fixed ellipsoidal boundary ^+^+^=1- ••• (1*) Substitotiiig in (4) we obtain (6»+c«)J={6«-c«)gr. (16) * Ic, ante p. 16. t It may be mentioned that, in the case of an incompressible fluid, equations somewhat similar to (4) had been established by Lagrange, MisctU, Taur. t. ix. (1760) [Oeuvrea, t. i. p. 442]. The author is indebted for this reference, and for the above remark on Helmholtz* investi- gation, to Sir J. Larmor. Equations equivalent to those given by Lagrange were obtained independently by Stokes, Ix. and made the basis of a rigorous proof of the velocity-potential theorem. % Nanson, Meae, of Math. t. viL p. 182 (1878). A similar interpretation of Helmholtz* equations was given by the author of this work in the Meas, of Math, t. vii. p. 41 (1877). Finally it may be noted that another proof of Lagrange's theorem, based on elementary dynamical principles, without special reference to the hydrokinetic equations, was indicated by Stokes, Comb, Trans, t, viii. [Papere, t. i p. 113], and carried out by Kelvin in his paper on Vortex Motion. S Cf. Voigt, "Beitrftge zur Hydrodynamik,'* Gott, Nachr, 1891, p. 71 ; Tedone, Nuovo CimetUo, t, xxxiiL (1893). The artifice in the text is taken from Poincar6, "Sur la precession dee corps d^ormables,*' BvU. Asir. 1910. 200 Vortex Motion [oh. vn whioh may be written a«(6« +c«) ^ = {6«(c«+a*) -c*(a*+6*)} qr, (17) with two similar equatiouB. We have here an identity as to fonn with Euler's equations of free motion of a solid about a fixed point. We easily deduce the integrals ^ + ?1 + ^ = const. (18) a* b^ c* ^ ' 6Vf* c«o«i;» a«6«f« and iT— H+-i — ~i+ • M =oonst, (19) 6"+c* e*+a* a*+b* ^ ' the former of which is a verification of one of Heknholtz' theorems, whilst the latter follows from the constancy of the energy. 147. It is easily seen by the same kind of argument as in Art. 41 that no continuous irrotational motion is possible in an incompressible fluid filling infinite space, and subject to the condition that the velocity vanishes at infinity. This leads at once to the following theorem : The motion of a fluid which fills infinite space, and is at rest at infinity, is determinate when we know the values of the expansion (0, say) and of the component vorticities ^, 77, ^, at all points of the region. For, if possible, let there be two sets of values, u^, Vi, Wi, and u^y v^y w^, of the component velocities, each satisfjdng the equations dx^dy^ dz ^' ^^^ throughout infinite space, and vanishing at infinity. The quantities m' = Wj — Wjj v' = V^ — Vj, w' = M^i — W2 will satisfy (1) and (2) with 6, f , >y, J = 0, and will vanish at infinity. Hence, in virtue of the result above stated, they will everywhere vanish, and there is only one possible motion satisfying the given conditions. In the same way we can shew that the motion of a fluid occupying any limited simply-connected region is determinate when we know the values of the expansion, and of the component vorticities, at every point of the region, and the value of the normal velocity at every point of the boundary. In the case of an n-ply-connected region we must add to the above data the values of the circulations in n several independent circuits of the region. 146-148] Velocities due to a Vortex-System 201 148. If, in the case of infinite space, the quantities 0, f , t;, ^ all vanish beyond some finite distance of the origin, the complete determination of UyVfW'wi terms of them can be effected as follows*. « The component velocities due to the expansion can be written down at once from Art. 56 (1), it being evident that the expansion 0' in an element Sx'Sy'Sz' is equivalent to a simple source of strength 0'Sx'8y'8z\ We thus obtain 9^ 34> 9^ ... ^"■"^' ""^"d^' ^^^^' ^^) where ' ^^^ fjjjdafdy'dz^, (2) r denoting the distance between the point {x'y y\ z') at which the volume- element of the integral is situate and the point {x, y, z) at which the values of u, V, w are required, viz. f = {(a? - xr + {y- y'Y + (2 - «')*}*, and the integration including all parts of space at which ff differs from zero. To find the velocities due to the vortices^ we note that when there is no expansion, the flux across any two open surfaces bounded by the same curve as edge will be the same, and will therefore be determined solely by the configuration of the edge. This suggests that the flux through any closed curve may be expressed as a line-integral taken round the curve, say i(Fdx + Ody + Hdz) (3) On this hypothesis we shall have, by the method of Art. 31, _dH dO dFdH _dO dF ^""9^ 9z' ^^dz dx' ^"^dx dy ^*^ To test the assumption, we must have 9t«^_9v_ 9_/aF; 9^ 9ff\ ^~9y dz'^dxKdx'^ dy^ dz) ^' with two similar equations. The quantities F, 0, H will in any case be indeterminate to the extent of three additive functions of the forms dx/dxy dx/dyy 9^/92;, respectively; and we may, if we please, suppose x to be chosen so that di + d^-^W^' (^) in which case ^*F = - ^, V«(?=-ij, V»^ = -J (6) * The investigation which follows is substantially that given by Heimholtz. The kine- matical problem in qnestion was first solved, in a slightly different manner, by Stokes, "On the Dynamical Theory of Diffraction," Camb. Trant. t. ix. (1849) [Paftrn, t iL pp. 2M. . .]. 202 Vortex Motion [oh. vn Paxticular solutions of these equations are obtained b;^ equating F, Gy H to the potentials of distributions of matter whose volume-densities are f/^Tr, >y/47r, C/^TT* respectively ; thus ^-^JIJ^T^'^tf'dz', O^ljlj^^dx'dy'dz', H = ljjj^;d.'d/dz', (7) where the accents attached to |, 17, ^ are used to distinguish the values of these quantities at the point (x\ y\ %'). The integrations are to include, of course, all places where ^, 17, J difEer from zero. ' Moreover, since 3/9a? . f-i s= — 3/3x' . r-^, the formulae (7) make The right-hand member vanishes, by the theorem of Art. 42 (4), since dx dy dz everywhere, whilst If H- m>y H- nj = at the surfaces of the vortices (where f, 17, ^ may be discontinuous), and ii Vy ^ vanish at infinity. Hence no additions to the values (7) of F, G, H are necessary in order that (5) may be satisfied. The complete solution of our problem is obtained by superposition of the results contained in (1) and (4), viz. dx dy dz ' dy dz dx 34) . dG dF (8) 325 dx dy ' where <!>, F, G, H have the values given in (2) and (7). It may be added that the proviso that d, f , r), J should vanish beyond a certain distance from the origin is not absolutely essential. It is sufficient if the data be such that the integrals in (2) and (7), when taken over infinite space, are convergent. This will certainly be the case if fl, f, ly, ^ are ultimately of the order jB-»», where R denotes distance from the origin, and n>3*. When the region occupied by the fluid is not unlimited, but is bounded (in whole or in part) by surfaces at which the normal velocity is given, and ♦ Cf. Leathern, Cambridge TracU, No. 1 (2nd ed.), p. 44. 148-149] Electro-magnetic Analogy 203 when further (in the case of an n-ply connected region) the value of the circulation in each of n independent circuits is prescribed, the problem may by a similar analysis be reduced to one of irrotational motion, of the kind considered in Chapter in., and there proved to be determinate. This may be left to the reader, with the remark that if the vortices traverse the region, beginning and ending on the boimdary, it is convenient to imagine them continued beyond it, or along the boundary, in such a manner that they form re-entrant filaments, and to make the integrals (7) refer to the complete system of vortices thus obtained. On this imderstanding the condition (5) will still be satisfied. There is an exact correspondence between the analytical relations above developed and those which obtain in the theory of Electro-magnetism. If, in the equations (1) and (2) of Art. 147, we write a, ft y, p, tt, v, w, p for u, V, w, B, f , 17, Cf ^f respectively, we obtain 9a 3/3 9y __ dx ^ dz ""'^' (9) 9y 9/3 __ 9a 9y _ 9j3 9a __ I 9y dz~ ' dz S~ ' ^ dy~ *) which are the fundamental relations of the theory referred to; viz. a, /3, y are the oolnpo- nents of magnetic force, «, v, U7 those of electric current, and p is the volume-density of the imaginary magnetic matter by which any magnetization present in the field may be repre- sented*. Hence, the vortex-filaments correspond to electric circuits, the strengths of the vortices to the strengths of the currents in these circuits, sources and sinks to positive and negative magnetic poles, and, finally, fluid velocity to magnetic force f. The analogy will of course extend to all results deduced from the fundamental relations ; thus, in equations (8), 4 corresponds to the magnetic potential and F,0,HU> the com- ponents of 'electro-magnetic momentum.' 149. To interpret the result contained in Art. 148 (8), we may calculate the values of u, v, w due to an isolated re-entrant vortex-filament situate in an infinite mass of incompressible fluid which is at rest at infinity. Since = 0, we shall have <E> = 0. Again, to calculate the values of Fy Gj Hy we may replace the volume-element Sx'Sy'Sz' by a'S*', where S*' is an element of the length of the filament, and a' its cross-section. Also f=a.^, ^=0.5^-., C=a.^. * Cf. Maxwell, SledrieUy and MagneHsm, Art. 607. The comparison has been simplified by the adoption of the 'rational* system of electrical units advocated by Heaviside, Eleetrical Papers, London, 1892, 1. 1 p. 199. t This analogy was first pointed out by Helmholtz; it has been extensively utilized by Kelvin in his papers on BleclraHaties and Magnetism (cited ante p. 37). 204 Vortex Motion [oh. vn frhere at' is the Torticity. Hence the formulae (7) of Art. 148 become ^-iirJT' ^-^]r' '^ 4,7.1 f' ^^' where #c, = co V, measures the strength of the vortex, and the integrals are to be taken along the whole length of the filament. Hence, by Art. 148 (4), we have wz\-*y')' with similar results for v, w. We thus find* _ K^ t/dy^ z- z' _ d^ y-^ \ ds^ ' **~4»rjl(fo' r ds' r )r*' — — {(^^' '" ~^' _ ^ g — g' \ <fe' ^~4^)\ds' r ds' r Jr>'} (2) If Aw, Av, ^w denote the parts of these expressions which involve the element ha' of the filament, it appears that the resultant of At^, Av, ^w is perpendicular to the plane containing the direction of the vortex-line at (x\ y\ z') and the line r, and that its sense is that in which the point (a?, y, z) would be carried if it were attached to a rigid body rotating with the fluid element at (a?', y', z'). For the magnitude of the resultant we have {(A«)« + (^vy + (A«;)«}» = ^ ^,^'. (3) where x is the angle which r makes with the vortex-line at {x\ y\ z'). With the change of symbols indicated in the preceding Art. this result becomes identical with the law of action of an electric current on a magnetic polef. Velocity -PoterUial due to a Vortex. 150. At points external to the vortices there exists a velocity-potential, whose value may be obtained as follows. Taking for shortness the case of a single re-entrant vortex, we have from the preceding Art., in the case of an incompressible fluid. «-£/(l';-*'-ii^') m * These are equivalent to the forms obtained by Stokes, Uc, arUe p. 201. t Ampere, TMorie mathinuUigue des phinomines ^ectro-dynamiquea, Paris, 1826. 149-150] Velocity-Potential due to a Vortex 205 By Stokes' Theorem (Art. 32 (4)) we can replace a line-integral extending round a closed curve by a surface-integral taken over any surface bounded by that curve ; viz. we have, with a slight change of notation, /,.^.w..^)=//{.(^-g)^».(g-g)-(g-|))^'. Bweput P-O, 0-1 !, Jt--^\. we find %' dz' dx'^r" dz' dx' dx'dy' r" dx' dy'~dx'dz'r" so that (1) may be written Hence, and by similar reasoning, we have, since d/dx' . r-^ = — d/dx . f-*, "=-S' ^=-^' «'="!' (2) where ^ = ^JJ(,|, + ,| + ,|,)1 ^' (3) Here I, w, n denote the direction-cosines of the normal to the element SS' of a surface bounded by the vortex-filament. The formula (3) may be otherwise written *-s(T ""JdS' (4) where ^ denotes the angle between r and the normal (Z, m, n). Since cos 6 SS'jr* measures the elementary solid angle subtended by SS' at (Xy y, z), we see that the velocity-potential at any point, due to a single re-entrant vortex, is equal to the product of K/4frr into the solid angle which a surface boimded by the vortex subtends at that point. Since this soUd angle changes by ^tt when the point in question describes a circuit embracing the vortex, we verify that the value of <f> given by (4) is cyclic, the cyclic constant being k. Cf. Art. 145. It may be noticed that the expression in (4) is equal to the flux (in the negative direction) through the aperture of the vortex, due to a point-source of strength k at the point {x, y, z). Comparing (4) with Art. 66 (4) we see that a vortex is, in a sense, equivalent to a uniform distribution of double sources over any surface bounded by it. The axes of the double sources must be supposed to be 206 Vortex Motion [ch. vn everywhere normal to the surface, and the density of the distribution to be equal to the strength of the vortex. It is here assumed that the relation between the positive direction of the normal and the positive direction of the axis of the vortex-filament is of the 'right-handed' type. See Art. 31. Conversely, it may be shewn that any distribution of double sources over a closed surface, the axes being directed along the normals, may be replaced by a system of closed vortex-filaments lying in the surface*. The same thing will appear independently from the investigation of the next Art. Vortex-Sheets. 151. We have so far assumed w, v, w to be continuous. We may now shew how cases where surfaces of discontinuity present themselves may be brought within the scope of our theorems. The case of a discontinuity in the normal velocity alone has already been treated in Art. 58. If w, v, w denote the component velocities on one side, and u\ v\ w' those on the other, it was found that the circumstances could be represented by imagining a distribution of simple sources, with surface-density l{u' ~ u) + m (v' — v) + n(w' — w), where {, m, n denote the direction-cosines of the normal drawn towards the side to which the accents refer. Let us next consider the case where the tangential velocity (only) is discontinuous, so that l(u' -u) + m{v' '-v)-\-n{w' -w) = (1) We will suppose that the lines of relative motion, which are defined by the differential equations dx dv dz = -^ =-7^^—, (2) are traced on the surface, and that the system of orthogonal trajectories to these lines is also drawn. Let PQ, P'Q' be linear elements drawn close to the surface, on the two sides, parallel to a line of the system (2), and let PP' and QQ' be normal to the surface and infinitely small in comparison with PQ or P'Q', The circulation in the circuit P'Q'QP will then be equal to (?' ~" i) PQi where q, q' denote the absolute velocities on the two sides. This is the same as if the position of the surface were occupied by an infinitely thin stratum of vortices, the orthogonal trajectories above-mentioned being the vortex-lines, and the vorticity cj and the (variable) thickness Sn of the stratum being connected by the relation (o8n =z q' -- q (3) * Of. Maxwell, Electricity and Magnetism, Arts. 486, 652. 150-151] Vortex-Sheets 207 The same result follows from a consideration of the discontinuities which occur in the values of u, v, w as determined by the formulae (4) and (7) of Art. 148, when we apply these to the case of a stratum of thickness 8n within which ^, Tjy £ are infinite, but so that ^8n, lySw, $8n are finite*. It was shewn in Arts. 147, 148 that any continuous motion of a fluid filling infinite space, and at rest at infinity, may be regarded as due to a suitable arrangement of sources and vortices distributed with finite density. We have now seen how by considerations of continuity we can pass to the case where the sources and vortices are distributed with infinite volume- density, but finite surface-density, over surfaces. In particular, we may take the case where the infinite fluid in question is incompressible, and is divided into two portions by a closed surface over which the normal velocity is continuous, but the tangential velocity discontinuous, as in Art. 68 (12). This is equivalent to a vortex-sheet; and we infer that every continuous irrotational motion, whether cyclic or not, of an incompressible substance occupying any region whatever, may be regarded as due to a certain distribution of vortices over the boundaries which separate it from the rest of infinite space. In the case of a region extending to infinity, the distri- bution is confined to the finite portion of the boundary, provided the fluid be at rest at infinity. This theorem is complemelitary to the results obtained in Art. 58. The foregoing conclusions may be illustrated by means of the results of Art. 01. Thus when a normal velocity 8^ was prescribed over the sphere r=a, the values of the velocity- potential for the internal and external space were found to be a /r\* « , . a /a^^'*-^ respectively. Hence if d« be the angle which a linear element drawn on the surface subtends at the centre, the relative velocity estimated in the direction of this element will be 2n + l d8^ n(n + l) 8c ' The resultant relative velocity is therefore tangential to the surface, and perpendicular to the contour lines {8f^=coDst.) of the surface-harmonic 8^, which are therefore the vortex- lines. For example, if we have a thin spherical shell filled with and surrounded by liquid, moving as in Art. 92 parallel to the axis of x, the motion of the fluid, whether internal or external, will be that due to a system of vortices arranged in parallel circles on the sphere ; the strength of an elementary vortex being proportional to the projection, on the axis of x, of the breadth of the corresponding zone of the surfacef. * Helmholtz, l,c. ante p. 194. t The same statements hold also for an ellipsoidal shell moving parallel to one of its principal axes. See Art. 114. 208 Vortex Motion [oh. vn Impulse and Energy of a Vortex-System. 152. The foUowmg investigations relate to the case of a vortex-system of finite dimensions in an incompressible fluid which fills infinite space and is at rest at infinity. The problem of finding a distribution of impulsive force (X\ Y\ Z') per unit mass which would generate the actual motion {u, v, w) instantaneously from rest is to some extent indeterminate, but a sufficient solution for our purpose may be obtained as follows. We imagine a simply-connected surface /S to be drawn enclosing all the vortices. We denote by <f> the single- valued velocity-potential which obtains outside Sy and by <f>i that solution of V^^ = which is finite throughout the interior of S, and is continuous with <f> at this surface.- In other words, (f>i is the velocity-potential of the motion which would be produced within S by the application of impulsive pressures fxf} over the surface. If we now assume X' = u + ^-K Y' = v + ^\ Z' = w + ^' (1) OX oy' dz ' at internal points, and z' = o, r' = o, z' = o : (2) at external points, it is evident on reference to Art. 11 that these forces would in fact generate the actual motion instantaneously from rest, the distribution of impulsive pressure being given by p<f> at external, and p<f>i at internal, points. The forces are discontinuous at the surface, but the discontinuity is only in the normal component, the tangential components vanishing just inside and just outside owing to the continuity of (f> with (f>i. Hence if (I, m, n) be the direction-cosines of the inward normal, we shall have mZ' -wr = 0, nZ'-JZ' = 0, lY'-mX' = 0, (3) at points just inside the surface. Now if we integrate over the volume enclosed by S we have jjjiyC - «?) ^dydz = jjj |y (I - I) - z g - g)]. dxdydz -/f/H^'-f)-'(f-f)l"»* ■= - /J{y (ly -mX')-z {nX' - IZ')} dS+2 ii^X' dxdydz, ... (4) where the suiface-integral vanishfis in virtue of (3). 152-153] Impulse of a Vortex-System 209 Again - ///(y* + ^») idxdydz^- jjfiy* + z«) (^ - I) dxdydz = -//^'-^^'Hf-^>'^^^^ ..(6) = my* + «•) (mZ'-nY')dS + 2 ///(yZ' - zY') dxiyiz, .... (5) where the surface-integral vanishes as before. We thus obtain for the force- and couple-resultants of the impulse of the Tortex-system the expressions ^ = iP iiSiyC -zri) dxdydz, i = - i/o ///(y* + 2«) $ dxdydz, Q = i/» iiSi^ - xO dxdydz, M = - i/) /;;(z» -I- X*) 7) dxdydz, R^'Ip ///(as? - yi) dxdydz, N = - ^p //J(a;» -|- y«) J <irdy dz. J To apply these to the case of a single re-entrant vortex-filament of infinitely small section a, we replace the volume element by oSs, and write J. dx dy y dz $ = o>-^, V = <^^, C=co^ (7) Hence P = ipcjo i(ydz — zdy) = Kp JJWS', (8) L = -ipaHTS(y^'\-z^)dx==''KpSS{mz^ny)d8\ (9) with similar formulae. The line-integrals are supposed to be taken along the filament, and the surface-integrals over a barrier bounded by it. The identities of the different forms follow from Stokes' Theorem. We have also written k for okt, i.e. k is the circulation round the filament*. The whole investigation has reference of course to the instantaneous state of the system, but it may be recalled that, when no extraneous forces act, the impulse is, by the argument of Art. 119, constant in every respect. 153. Let us next consider the energy of the vortex-system. It is easily proved that under the circumstances presupposed, and in the absence of extraneous forces, this energy will be constant. For if T be the energy of the fluid bounded by any closed surface S, we have, putting F = in Art. 10 (5), DT -jT- = ff{lu 4- wv -f nw) pdS (1) * The ezprefisions (8) and (9) were obtained by elementary reasoning by J. J. Thomson, On the Motion of Vortex Rings (Adams Prize Essay), London, 1883, pp. 5, 6, and the formulae (6) deduced from them, with the opposite signs, however, in the case of L, M, N, The correction is due to Mr Welsh. An interesting test of the formnlae as they now stand i» afforded by the case of a spherical mass rotating as if solid and surrounded by fluid at rest, provided we take into account the spherical vortex-sheet which represents the discontinuity of velocity. L. H. 14 210 Vortex Motion [chap, vn If the surface S enclose all the vortices, we may put E.|_l^H.,(0 ,2, and it easily follows from Art. 150 (4) that at a great distance R from the vortices p will be finite, and lu -{- mv -{- nw of the order fi-', whilst when the surfacQ S is taken wholly at infinity, the elements 8S vary as iZ*. Hence, ultimately, the right-hand side of (1) vanishes, and we have r = const (3) We proceed to investigate one or two important kinematical expressions for T, still confining ourselves, for simplicity, to the case where the fluid (supposed incompressible) extends to infinity, and is at rest there, all the vortices being within a finite distance of the origin. The first of these expressions is indicated by the electro-magnetic analogy pointed out in Art. 148. Since 6 = 0, and therefore <E> — 0, we have 22* = /) JJJ(m« + v* + w*) dxdydz by Art. 148 (4). The last member may be replaced by the sum of a surface- integral p ll{F (mw — nv) -\- [nu — Jw?) 4- i? (fc — rnu)} dS, and a volume-integral At points of the infinitely distant boundary, F, 0, H are ultimately of the order R-\ and w, v, w of the order fi-*, so that the surface-integral vanishes, and we have T = ipJ/K^I + Gr, + Hi;) dxdydz, (4) or, substituting the values of F, 0, H bom. Art. 148 (7), ^ " iirlJIIJI^^' ^ ^r^ ^ - dxdydzdx'du'dz', (5) where each volume-integration extends over the whole space occupied by the vortices. A slightly different form may be given to this expression as follows. Regarding the vortex-system as made up of filaments, let Ss, Ss' be elements of length of any two filaments, a, a the corresponding cross-sections, and w, iii the corresponding vorticities. The elements of volume may be taken j to be ahs and a'S^', respectively, so that the expression following the integral signs in (5) is equivalent to cos € 5 , ,5 , . ii}Q08 , <JJ a OS , 153 j Energy of a Vortex-System 211 where c is the angle between 8* and S«'. If we put cja == #c, coV = #c', we have T^;l^JlKK'jj^ dads', (6) where the double integral is to be taken along the axes of the filaments, and the summation S includes (once only) every pair of filaments which are present. The faotor of p in (6) is identioal with the expression for the energy of a system of electric currents flowing along oonduotors coincident in position with the vortex-filaments, with strengths k, k, ... respectively*. The above investigation is in fact merely an inversion of the argument given in treatises on Electro-magnetism, whereby it is proved that ^ 2 it'll ^ dads' =i III (a* +/8« -h /) dxdydz, 4, i' denoting the strengths of the currents in the linear conductors whose elements are denoted by bs, ha', and a, /3, y the components of magnetic force at any point of the field. The theorem of this Art. is purely kinematical, and rests solely on the assumption that the functions u, v, w satisfy the equation of continuity, dx dy dz" ' throughout infinite space, and vanish at infinity. It can therefore by an easy generaliza- tion be extended to a case considered in Art. 144, where a liquid is supposed to circulate irrotationally through apertures in fixed solids, the values of u, v, w being now taken to be zero at all points of space not occupied by the fluid. The investigation of Art 151 shews that the distribution of velocity thus obtained may be regarded as due to a system of vortex-sheets coincident with the bounding surfaces. The energy of this system will be given by an obvious adaptation of the formula (6) above, and will therefore be proportional to that of the corresponding system of electric current-sheets. This proves a statement made by anticipation in Art. 144. Under the circumstances stated at the beginning of Art. 152, we have another usqjhil expression for T; viz. ^ T = piSi{u{yC-zri) + v{zi-xO-{-w(xri-yi)}dxdydz1[. ...(7) To verify this, we take the right-hand member, and transform it by the process already so often employed, omitting the surface-integrals for the same reason as in the preceding Art. The first of the three terms gives '' ///" HI - 1) - Ki^ - ©} ^""^y^' = — pl\l\(vy -{-tjoz)-^ w* ■ dxdydz. * The 'rational' system of electrical units being understood; see ante p. 203. t Motion of Fluids, Art. 136 (1879). 14—2 212 Vortex Motion [chap, vn Trausf orming the remaining terms in the same way, adding, and making use of the equation of continuity, we obtain or, finally, on again transforming the last three terms, ip SSR'^^ + v^ + w^) dxdydz. In the case of a finite region the surface-integrals must be retained*. This involves the addition to the right-hand side of (7) of the term P JJ{(^ -h wv -h nw) {xu + yv -{- zw) — J (Za; -f my -h nz) g*} dS, ... (8) where g* = w* -h v* -h wK This simplifies in the case of affixed boundary. The value of the expression (7) must be unaltered by any displacement of the origin of co-ordinates. Hence we must have /JJ(vJ — ^) dxdydz = 0, i!i(wi — u^) dxdydz = 0, //J(wiy — t?^) dxdydz = 0, (9) These equations, which may easily be verified by partial integration, foUow also from the consideration that if there are no extraneous forces the components of the impulse parallel to the co-ordinate axes mu9t be constant Thus, taking first the case of a fluid enclosed in a fixed envelope of finite size, we have, in the notation of Art. 152, P = pjjjudxdydz -p J/^<Mf, (10) if ^ denote the velocity-potential near the envelope, where the motion is irrotationaL Hence ^=p j j j^dxdydz-p j jr^dS = -P j j j^^dxdydz+p j j j (v(-wn)dxdydz-pjjl^dS, (11) by Art. 146 (5). The first and third terms of this cancel, since at the envelope we have X =^l^f ^y '^'t. 20 (4) and Art. 146 (6). Hence for any re-entrant system of vortioea enclosed in a fixed vessel, we have -^ =P Hi W -WTiy) dxdydz, (12) with two similar equations. It ha.s been proved in Art. 119 that if the containing vessel be infinitely large, and infinitely distant from the vortices, P is constant. This gives the first of equations (9). Conversely from (9), established otherwise, we could infer the constancy of the com- ponents P, Q, R of the impulse*. * J. J. Thomson, he. ante p. 209. 153-154J Two-Dimensional Theory 213 RectiJinea/r Vortices. 154. When the motion is in two dimensions cr, y we have w ^Q, whilst t*, V are functions of cr, y, only. Hence f = 0, ly = 0, so that the vortex-lines are straight lines parallel to z. The theory then takes a very simple form. The formulae (8) of Art. 148 are now replaced by ~" dx dy' ~ dy dx' ' the functions <f>, tff being subject to the equations V,V-~fl, V,V-f, (2) and to the proper boundary-conditions. In the case of an incompressible fluid, to which we wiU now confine our- selves, we have »-i- «-i. ") where ^ is the stream-function of Art. 59. It is known from the theory of Attractions that the solution of ViV=^. (4) i being a given function of Xy y, is 1^ = ^//riogr<fe'dy' + i^o. (5) where £' denotes the value of ^ at the point {x\ y% and r stands for {(X - x^)* + (y- y')«}*. The 'complementary function' ^o '^^y ^® ^^7 solution of Vl^o = 0; (6) it enables us to satisfy the boundary-conditions. In the case of an unlimited mass of Uquid, at rest at infinity, ^o ^ ^^^' stant. The formulae (3) and (5) then give Hence a vortex-filament whose co-ordinates are x', y' and whose strength is k contribates to the motion at (z, y) a velocity whose components are _ K_ y-y; , K_ x-x' 214 Vortex Motion [chap, vn This velocity is perpendicular to the line joining the points (a, y), (x\ y% and its amount is /c/277r. Let us calculate the integrals iiuldxdy, and Hv^dxdyy where the integra- tions include all portions of the plane ay for which f does not vanish. We have Iju^dxdy = - ^jjjja'y^ dxdydx'dy', where each double integration includes the sections of all the vortices. Now, corresponding to any term a'^-^dxdycbfdy' of this result, we have another ^'l^dxdydx'dy\ and these two neutralize each other. Hence, and by similar reasoning, jiu^dxdy = 0, Sfv^dxdy = (8) If as before we denote the strength of a vortex by #c, these results may be written S/m = 0, S#rt; = (9) Since the strength of each vortex is constant with regard to the time, the equations (9) express that the point whose co-ordinates are ^^-=27' y-S (^^) is fixed throughout the motion. This point, which coincides with the centre of inertia of a film of matter distributed over the plane xy with the surface-density J, may be called the 'centre' of the system of vortices, and the straight line parallel to z of which it is the projection may be called the 'axis' of the system. If Zic = 0, the centre is at infinity, or else indeterminate. 155. Some interesting examples are furnished by the case of one or more isolated vortices of infinitely small section. Thus : 1°. Let us suppose that we have only one vortex-filament present, and that the vorticity ^ has the same sign throughout its infinitely small section. Its centre, as just defined, will lie either within the substance of the filament, or infinitely close to it. Since this centre remains at rest, the filament as a whole will be stationary, though its parts may experience relative motions, and its centre will not necessarily lie always in the same element of fluid. Any particle at a finite distance r from the centre of the filament will describe a circle about the latter as axis, with constant velocity 154-156] Vortex-Pair 215 KJimT, The region external to the vortex is doubly-connected; and the circulation in any (simple) circuit embracing it is of course k. The irrotational motion of the surrounding fluid is the same as in Art. 27 (2)« 2°. Next suppose that we have two vortices, of strengths /Ci, /Cj, respec- tively. Let A^ B be their centres, the centre of the system. The motion of each filament as a whole is entirely due to the other, and is therefore always perpendicular to AB. Hence the two filaments remain always at the same distance from one another, and rotate with constant angular velocity about 0, which is fixed. This angular velocity is easily found; we have only to divide the velocity of A (say), viz. /C2/(27r . AB), by the distance -40, where A0 = Ki 4- /Cj AB, and so obtain Ki + K^ 2n.AB^' If #ci, K^ be of the same sign, i.e. if the directions of rotation in the two vortices be the same, lies between A and B; but if the rotations be of opposite signs, Ues in AB, or BA, produced. If /Ci = — /Cj, is at infinity; but it is easily seen that A, B move with equal velocities /Ci/(27r . AB) at right angles to AB, which remains fixed in direction. Such a combination of two equal and opposite vortices may be called a 'vortex-pair.' It is the two-dimensional analogue of a circular vortex-ring (Art. 160), and exhibits many of the properties of the latter. The stream-lines of a vortex-pair form a system of coaxal circles, as shewn on p. 68, the vortices being at the limiting points {± a, 0). To find the 216 Vortex Motion [chap, vn relative stream-lines, we superpose a general velocity equal and opposite to that of the vortices, and obtain, for the relative stream-function, -&te+>»*r;) m in the notation of Art. 64, 2^. The figure (which is turned through 90° for convenience) shews a few of the lines. The line ^ =: consists partly of the axis of y, and partly of an oval surrounding both vortices. It is plain that the particidar portion of fluid enclosed within this oval accompanies the vortex-pair in its career, the motion at external points being exactly that which would be produced by a rigid cylinder having the same boundary; cf. Art. 71. The semi-axes of the oval are 2*09 a and 1*73 a, approximately*. A difficulty is sometimes felt, in this as in the analogous instance of a vortex-ring, in understanding why the vortices should not be stationary. If in the figure on p. 68 the filaments were replaced by solid cylinders of small circular section, the latter might indeed remain at rest, provided they were rigidly connected by some contrivance which did not interfere with the motion of the fluid; but in the absence of such a connection they would in the first instance be attracted towards one another, on the principle explained in Art. 23. This attraction is however neutralized if we superpose a general velocity V of suitable amount in the direction opposite to the cyclic motion half-way between the cylinders. To find F, we remark that the fluid velocities at the two points (a ±^ c, 0), where c is small, will be approximately equal in absolute magnitude, provided V '\-— ^ =_!L +_!! V 2irC 4)ra 2irC ^na ' where k is the circulation. Hence F = 47ra' which is exactly the velocity of translation of the vortex-pair, in the original form of the problem t* Since the velocity of the fluid at all points of the plane of symmetry is wholly tangential, we may suppose this plane to form a rigid boundary of the fluid on either side of it, and so obtain the case of a single rectilinear vortex in the neighbourhood of a fixed plane wall to which it is parallel. The filament moves parallel to the plane with the velocity KJifirh, where h is the distance from the wall. Again, since the stream-lines are circles, we can also derive the solution of the case where we have a single vortex-filament in a space bounded, either internally or externally, by a fixed circular cyUnder. ♦ Cf. Sir W. Thomaon, "On Vortex Atoms," PhU. Mag, (4), t. xxxiv. p. 20 (1867) [PaperB, t. iv. p. 1]; and Biecke, Q6U, Nachr. 1888, where paths of fluid particles are also delineated. t A more exact investigation is given by Hicks, "On the Condition of Steady Motion of Two Cylinders in a Fluid," Quart. Jaum, Maih. t. xvii. p. 194 (1881X 155] Method of Images 217 Thus, in the figure, let EPD be the section of the cylinder, A the position of the vortex (supposed in this case external), and let B be the * image' of A with respect to the circle EPD, viz. C being the centre, let CB . CA =c«, — ^ where c is the radius of the circle. If P be any point on the circle, we have AP AE AD BP^m^BD^'^'^^''^ so that the circle occupies the position of a stream-line due to a vortex-pair at A, B, Since the motion of the vortex A would be perpendicular to AB, it is plain that all the conditions of the problem will be satisfied if we suppose A to describe a circle about the axis of the cylinder with the constant velocity K __ K . CA ' 2w . AB" ^ 2w {CA^ 'C*y where k denotes the strength of A, In the same way a single vortex of strength k, situated inside a fixed circular cylinder, say at B, would describe a circle with constant velocity K,CB It is to be noticed, however*, that in the case of the external vortex the motion is not completely determinate unless, in addition to the strength k, the value of the circulation in a circuit embracing the cylinder (but not the vortex) is prescribed. In the above solution, this circulation is that due to the vortex-image at B and is - k. This may be annulled by the superposition of an additional vortex + ic at (7, in which case we have, for the velocity of A, ,CA kc^ For a prescribed circulation k we must add to this the term ic72fr . CA, 3^. If we have four parallel rectilinear vortices whose centres form a rectangle ABB'A\ the strengths being #c for the vortices A\ S, and — k for the vortices A^ B\ it is evident that the centres will always form a rectangle. Further, the various rotations having the directions indicated in the figure * F. A. Tarieton, "On a Problem in Vortex Motion," Proc B, I. A. December 12, 1892. 218 Vortex Motion [chap, vn we see that the e£Eect of the presence of the pair A, A' onB, ff iBto separate them, and at the same time to diminish their velocity perpendicular to the line joining them. The planes which bisect AB, AA' at right angles may (either or both) be taken as fixed rigid boundaries. We thus get the case where a pair of vortices, of equal and opposite strengths, move towards (or from) a plane wall, or where a single vortex moves in the angle between two perpendicular waUs. If ;i:, ^ be the co-ordinates of the vortex A relative to the planes of symmetry, we readily find • - * ^* • - * y* /o\ ^-"i^P' ^-Tn'^' ^^^ where r^=x^+y^. By division we obtain the differential equation of the path, viz. dx dy whence o* (a;* + y*) = 4a:* y*, a being an arbitrary constant, or, transforming to polar co-ordinates, r=-T^ (3) Also since o:y -yx^— , the vortex moves as if under a centre of force at the origin. This force is repulsive, and its law is that of the inverse cube*. 156. If we write, as in Chapter iv., 2 = 35 + iy, U7 = <^ 4- i^, (1) the potential- and stream-functions due to an infinite row of equidistant vortices, each of strength #c, whose co-ordinates are (0, 0), (± a, 0), (± 2a, 0), . . . , will be given by the formula t£; = 2;^logsm-; (2) cf. Art. 64, 3"*. This makes dw tK ^ rrz .«. w - tv = - -V- = - s- cot — , (3) dz 2a a • Greenhill, "On Plane Vortex-Motion," Quart. Joum. Math, t. xv. (1887); Grobli, Die Bewegung paraUeler geradliniger Wirhdfdden, Zurich, 1877. These papers contain other in- teresting examples of rectilinear vortex-Bystems. The case of a system of equal and parallel vortices whose intersections with the plane xy are the angular points of a regular polygon was treated by J. J. Thomson in his Motion of Vortex Rings, pp. $f4.... He finds that the configuration is stable if, and only if, the number of vortices does not exceed six. For some further references as to special problems see Hicks, BriL Ass, Rep. 18)^2, pp. 41 ... ; Love, Ix, anJte p. 183. An ingenious method of transforming plane problems in vortex -motion was given by Routh» '*Some Applications of Conjugate Functions," Proc. Lond. Math. Soe. t. xi]> p. 73 (1881). 155-166] Rows of Vortices 219 whence U^ — TT- sinh {^tryja) V^TT- sin (2mxla) 2a cosh {2ny/a) — cob {2nrx/a) ' " 2a cosh {2ny/a) — cos (2w«/a) ' (4) These expressions make w = =f J /c/a, v = 0, f or y = ± oo ; the row of vortices is in fact, as regards distant points, equivalent to a vortex-sheet of uniform strength #c/a (Art. 161). The diagram shews the arrangement of the stream-lines. It follows easily that if there are two parallel rows of equidistant vortices, symmetrical with respect to the plane y = 0, the strengths being #c for the upper and — #c for the lower row, as indicated on the next page, the whole system will advance with a uniform velocity U^^coth-, (5) 2a a ^ ' where b is the distance between the two rows. The mean velocity in the plane of symmetry is /c/a. The velocity at a distance outside the two rows tends to the Umit 0. If the arrangement be modified so that each vortex in one row is opposite the centre of the interval between two consecutive vortices in the other row, as shewn on p. 222, the general velocity of advance is 2a a (6) The mean velocity in the medial plane is again x/a. The stability of these various arrangements has been discussed by von K&rm4n*. Taking first the case of the single row, let us suppose the vortex whose undisturbed co-ordinates are {ma, 0) to be displaced to the point {ma +Xm» ^m)* ^o formulae of Art. 154 give (7) ^0 _ « « yp-ym where 2ff m *'m rn? = (a^o - a:„, - ma)^ + {y^ - yJi\ (8) * "FIuBsigkeits- u. Luftwiderstand," Phya, Zeitachr, t. xiii (1912) ; also Om, Nachr. 1912, p. 547. The investigation is only given in outline in these papers; I have supplied various steps. 220 Vortex Motion [chap, vn and the summation with respect to m inoludes all positive and negative integral values, zero being of ooorse excluded. If we neglect terms of the second order in the displacements, we find The first term in the value of dyo/dt is to be omitted as being independent of the disturbance*. Consider now a disturbance of the type x«=ae**~*, yn.=/3e**^, (10) where <f> may be assumed to lie between and 2ir. If ^ be small this has the character of a wave of length 27ra/^. We find |=-Xft f=-X„ (11) , X K /1-C08d> 1 -COS 20 l-cos3d> \ « , ,« .. ,,«v where X=— ,( ^ ^, ^ + ^T-^ + 3«-^ + ' * 'j =£?*(2ir -0). ..(12) The arrangement is therefore unstable, the disturbance ultimately increasing as e^^ When the wave-length is large comi)ared with a we have X =iK<f>la*, (13) approximately; cf. Art. 234. Proceeding next to the case of the symmetrical double row, the positions at time t of vortices in the upper and lower rows may be taken to be (wia + Ut + x^, \h + ym), and (wa + C7« + re/, - ^6 + y/)* respectively, where TJ denotes the general velocity of advance of the system, and the origin is in the plane of symmetry. <3) c?) <5) 6) (J (J) <j) (J The component velocities of a vortex in the upper row, e.g, that for which m = 0, due to the remaining vortices of the same row, will be given as before by (9), where the sum 1m~^ may be omitted. The components due to the vortex n of the lower row will be 2n r„« ' " 27r f^* where r„« = (xo - Xn - na)^ + (yo - y«' + b)K If we n^lect terms of the second order in the disturbance we find, after a little reduction, 2iuib ^!(^?5^TT»T«^'^^-'^'*'> ^^^^ K (ft - " « m2a« *** n (w*a« + 6«)« ^^^^ " ^" ^ * In the summations the vortices are to be taken in pairs equidistant from the origin ; otherwise the result would be indeterminate. The investigation may be regarded as applying to the central portions of a long, bat not infinitely long, row; the term referred to is then negligible. 156] Stability of Double Mows 221 where the summations with respect to n go from - ao to + oo , including zero. The terms in (14) independent of the disturbance will cancel, since, by (5) C7= 5-COth — = gr-2 a , . ., . If we now put ar,^=ae'«*, ym = /3e'«*, ar/ = a'e"^, y/ = i8 V^ (16) where < ^ < 2ir, the equations take the form ?!«!^=_^„ -Ca' + ^/s-.l If we write, for shortness, k = b/a, (18) the values of the coefficients are* ^ = 1^?-- !(-^?Ti.). = 4*<2.-*) + 55g^ (19) _ ^ 2nke*^ _ fir<^ cosh fc (w - <^) g« sinh l;<f) l »(n« + ifc*)«" 1 sinhifcir " sinh'ijir J ' ^^"^ (n« - l^) e*"^ _ ir» cosh k<l> ff<^ sinh ifc (w - <^) ^^n (»« + ik*)« """ sinh«ifcir " sinhifeir ^^^^ To deduce the equations relating to the lower row we have merely to reverse the signs of K and 6, and to interchange accented and unaccented letters. Hence " ^' \ (22) ^ = -4a' + Ca + Bfi. K at J The formulae (17) and (22) are the equations of motion of the vortex-system in what may be called a normal mode of the disturbance. The solutions are of two types. In the first type we have a^a\ i8= -i8' (23) and therefore * I (24) The solution involves exponentials e^^ the values of X being given by ^^X= 'B±J(A^-C^) .' (26) K * The summations with respect to n can be derived from the Fourier expansion oo8hifc(T~0) _ 1 jl 21; COB » 21; COB 2^ 1 Bmhl;ir "«■ t* 1*+** "*" 2»+i* +•••[• 222 Vortex Motion [chap, vn In the seoond type we have a^-a', /3=/3' (26) and therefore -^ jt = Ba - (A + C)ff, ] ' * I (27) ___=_(^.C)a+mJ The corresponding values of X are given by ?^ X = B±^(A* - C«) (28) Since B is a pure imaginary, whilst A and C are real, it is necessary for stability in each case that A* should not exceed C for admissible values of (f>. Now when ^ = n- we find u4 + C = Jw* tanh« Jifcir, A - C = in* coth« ikir, (29) so that A* - C^iB positive. We conclude that both types are unstable. Passing to the unsymmetrical case, we denote the positions of the displaced vortices by {ma + F« + Xn,f ib + y^), and ((n + i)a -\- Vt + Zn, - J6 + yn), where F is given by (6). The requisite formulae are obtained by writing n + ^ f or n in preceding results. <J <3) <3) <5) <3) ^ ^ The equations (17) and (22) will accordingly apply, provided* 1 - e**^ ^ (w + i)* - it« - . ,- ^, «•« {2n+ 1) fce*(»+*)^ _ . f 7r<^8inhA;(7r-<f>) n-^sinh^l » {(n + f)« + ^»}« ""*( cosh^TT "*" coshH-irJ' ^^^^ p , ((w + i)' - ^} e^ ^"""^^ * _ ir« cosh A;<» ir<<) cosh fe (,r - ») » {(w + i)* + ^}* "" oo8h« kn ' cosh ifctr ^"^^^ These values of ^, JS, C are to be substituted in (25) and (28). As in the former case it is necessary for stability that A* should not be greater than C*. Now when ^ = ir, C = 0; hence A must also vanish, or C08h«ifcir = 2, ifcn- = -8814, 6/o=ifc = -281 (33) The configuration is therefore unstable unless the ratio of the interval between the two rows to the distance between consecutive vortices has precisely this value. To determine whether the arrangement is stable, under the above condition, for all values of ^ from to 2Yr, let us write for a moment k(ir - <t)) = x, kv = fi, bo that J^A = - ioc*, k^C = i(fix cosh fix cosh a; - /n* sinh fj, sinh x) ( 34) * The summations with respect to n can be derived from the ezpansion sinh A; (t - 0) _ 2 \k cos i<p , k cos |^ cosh ibr 2 \ k cos i<p k cos 1^ , j "i^ l(i)* + it«+(i)rnbS + ---r 166-157] Stability of Double Mows 223 where x may range between ±fi. Since ^ is an even and C an odd function of a;, it is sufficient for comparison of absolute values to suppose x positive. Hence, writing y=fi cosh fi cosh z - fi^ sinh /n x, .(36) X we have to ascertain whether this is positive for <x </i. Since /li = •8814, cosh /x = ^2, sinh fi= If yjs positive for x = 0, and it evidently vanishes for x= /a. Again dy , • u . « • u sinh a; , . , coshx , ,«^, ^ = /i cosh fi smh « + /i* smh fi — j /i* smh/x 1, (36) which is equal to - 1 f or x = 0, and vanishes toTX = fi. Finally, tPy , , • • u sinh a; ^ - . , cosh a? « • . , sinh a; ,„_. ;t4 = /* cosh /i cosh X - /Li* sinh ;i- — + 2/Li* sinh /i — -^-z 2/i' smh /a , , . .(37) (M/ a? X a» which is easily seen to be positive for all values of x, since (tanha;)/a;<l. Hence as x increases from to /^ dy/dx is steadily increasing from - 1 to 0, and is therefore n^ative. Hence y steadily diminishes from its initial positive value to zero, and is therefore positive. We conclude that the configuration is definitely stable* except for a: = ± /i, when (^ = or 2ir, in which cases B = 0, by (31), and therefore X = 0. Since the disturbed particles are then all in the same phase, the reason why the period of disturbance should be infinite is easily perceived. ■ 157. When, as in the case of a vortex-pair, or a system of vortex-pairs, the algebraic sum of the strengths of all the vortices is zero, we may work out a theory of the impulse,' in two dimensions, analogous to that given in Arts. 119, 152 for the case of a finite vortex-system. The detailed exami- nation of this must be left to the reader. If P, Q denote the components of the impulse parallel to x and y, and N its moment about Oz, all reckoned per unit depth of the fluid parallel to z, it will be found that P = pny^dxdy, Q = -pSSxCdxdyA . N^--\pll{x^ + y^)ldxdy. J For instance, in the case of a single vortex-pair, the strengths of the two vortices being ± k, and their distance apart c, the impulse is pKC, in a line bisecting c at right angles. j (2) The constancy of the impulse gives ^KX = const., J^Ky = const., S/c (x^ + y^) = const. It may also be shewn that the energy of the motion in the present case is given by T = ^yss^^dxdy=-y:s:K^ (3) When Sic is not zero, the energy and the moment of the impulse are both infinite, as may be easily verified in the case of a single rectilinear vortex. * This is stated without proof by KdrmiLn. (4) 224 Vortex Motion [chap, vn The theory of a system of isolated rectilinear vortices has been put in a very elegant form by Kirchhoff*. Denoting the positions of the centres of the respective vortices by (z^, yi)f (x^» y^iy . . . and their strengths by k^, k,, . . . , it is evident from Art. 154 that we may write da^_ W dy^ dW "^dt^'dy^* ""(ft'ax, ' where W =^ 2*, «, log r„, .., (6) if fit denote the distance between the vortices k^, k^. Since W depends only on the rdaiive configuration of the vortices, its value is unaltered when Xi, x^, ... are increased by the same amount, whence id W/dxi =0, and, in the same way, 2dW/dyi=0, This gives the first two of equations (2), but the proof is not now limited to the case of 2k =0. The argument is in fact substantially the same as in Art. 154. Again, we obtain from (4) ^ / dx dy\ ^/ dW dW\ or if we introduce polar co-ordinates (r^, ^i), (r,, B^), ... for the several vortices, ^"'S^'^^ (^> Since W is unaltered by a rotation of the axes of co-ordinates in their own plane about the origin, we have tdW/dO =0, whence 2«cr*=const., (7) which agrees with the third of equations (2), but is free from the restriction there implied. An additional integral of (4) is obtained as follows. We have ^ f dy dx\ ^/ dW dW\ or ^""^dt^^^'W (®) If every r be increased in the ratio 1 +c, where c is infinitesimal, the increment of IF is equal to Scr . dW/dr, But since the new configuration of the vortex-system is geometrically similar to the former one, the mutual distances r^ are altered in the same ratio 1 +f, and therefore, from (5), the increment of IF is f/2n- . Ik^k^, Hence (8) may be written in the form ^'''^di^2ir^''^''' (^) 158. The preceding results are independent of the form of the sections of the vortices, so long as the dimensions of these sections are small compared with the mutual distances of the vortices themselves. The simplest case is of course when the sections are circular, and it is of interest to inquire whether this form is stable. This question has been examined by Kelvin f. * Jiechanik, c. zx. t Sir W. Thomson, "On the Vibrations of a Columnar Vortex," Phil Mag. (6), t. x. p. 156 (1880) [Papers, t. iv. p. 152]. 167-158] Stability of a Cylindrical Vortex 225 WEen'the distorbaiioe is in two dimensions only, the calculations are veiy Simple.. Let us suppose, as in Art. 27, that the space within a circle r—a, having the centre as origin, is occupied by fluid having a uniform vorticity o>, and that this is surrounded by fluid moving irrotationally. If the motion be continuous at this circle we iiave, for r <.ay ^=-J«(a*-r«) (1) while for r > a, ^ = - Jwa* log a/r. (2) To examine the effect of a slight irrotational disturbance, we assume, for r K^a, ■^ = - J« (o* -r*) +A — 008 (t6 - <r<).] " I (3) and, for r> a, ^= -^a*log -+-4 — cos(«^-aO>j where 9 is integral, and o- is to be determined. The constant A must have the same value in these two expressions, since the radial component of the velocity, -d^/rd^, must be continuous at the boundary of the vortex, for which r^a^ approximately. Assuming for the equation to this boundary r =a +a cos («^ -crO. (4) we have still to express that the transverse component (d^/Br) of the velocity is continuous. This gives Substituting from (4), and neglecting the square of a, we find wi^-ZaA/a (5) So far the work is purely kinematical; the dynamical theorem that the vortex-lines move with the fluid shews that the normal velocity of a particle on the boundary must be equal to that of the boundary itself. This condition gives 8r_ 9^ dyjt dr where r has the value (4), or A . 8a ,^. <ra=« — +4<»a. — (o) Eliminating the ratio A/a between (5) and (6) we find <7=J(«-l)a) (7) Hence the disturbance represented by the plane harmonics in (3) consists of a system of corrugations travelling round the circumference of the vortex with an angular velocity <r/«= (« -!)/«. Jft) (8) This is the angular velocity in space; relative to the rotating fluid the angular velocity is v/8-^= -i»l8, (9) the direction being opposite to that of the rotation. When 9=2, the disturbed section is an ellipse which rotates about its centre with angular velocity ^«». The transverse and longitudinal oscillations of an isolated rectilinear vortex-filament have also been discussed by Kelvin in the paper cited. I1.H. 16 226 Vortex Motion [chap, vn 159. The particular caae of an elliptic disturbance can be solved without approximation as follows*. Let us suppose that the space within the ellipse ?!+y"=i (1) is occupied by liquid having a uniform vorticity a>, whilst the surrounding fluid is moving irrotationally. It will appear that the conditions of the problem can all be satisfied if we imagine the elliptic boundary to rotate, without change of shape, with a constant angular velocity (n, say), to be determined. The formula for the external space can be at once written down from Art. 72, 4^ ; viz. we have ^ = Jw (a +6)« t-^ cos 2»7 + JcDO^jf (2) where (, 17 now denote the elliptic co-ordinates of Art. 71, 3°, and the cyclic constant k has been put =9ra6a). The value of ^ for the internal space has to satisfy 3a:* 3y^ with the boundary-condition -7 + rj = - wy . -^ +na; . ^ (4) These conditions are both fulfilled by ^ = J« (At^ ^By^) (6) provided A +JB = 1, Aa^ -JB6«=-(a« -6*) (6) It remains to express that there is no tangential slipping at the boundary of the vortex; ».e. that the values of 3^/8$ obtained from (2) and (5) there coincide. Putting x=c cosh f cos f)ty=c sinh f sin 17, where c — J(a*- 6*), differentiating, and equating coeffi- cients of cos 2i7, we obtain the additional condition -in (a +6)* e"*^ =\(ac^ (A -B) cosh J sinh f, where ( is the parameter of the ellipse (1). This is equivalent to .^-B=-».?!^* (7) since, at points of the ellipse, cosh ( =alCt sinh ( = 6/c. Combined with (6) this gives . Aa -Bh = — 7, (8) •"■^ "=(^«» <®> When a =6, this agrees with our former approximate result. The component velocities x, y oi a particle of the vortex relative to the principal axes of the ellipse are given by x= - ^ +ny, y=^'-nz, cy " '^ ex * KirchhoiT, Mechanik, c. xz. p. 261; Basset, Hydrodynamics, t. ii. p. 41. »• . 169.-161] Elliptic Vortex 227 j» V V X whence we find -=-n|, f = »-^ (10) a o a Int^^ting, we find a; =fai cos (rrf + c)^ y==tt8in(n<+«), (11) where A;, c are arbitrary constants, so that the relative paths of the particles are ellipses similar to the boundary of the vortex, described according to the harmonic law. If x', f/ be the co-oidinates relative to axes fixed in space, we find x' =x oos fU-ymnnt=ik{a+b) cos (2nt + c) + ^ib (a - b) cos c. + f)+ih{a-b)<iOBtA + <) -ik{a-b) sine./ The absolute paths are therefore circles described with angular velocity 2n*. (12) i/^xwBtU+y cos tU =^k (a +6) sin {2rU 160. It was pointed out in Art. 80 that the motion of an incompressible fluid in a curved stratum of small and imif orm thickness is completely defined by a stream-function iff, so that any kinematical problem of this kind may be transformed by projection into one relating to a plane stratum. If, further, the projection be *orthomorphic,' the kinetic energy of corresponding portions of liquid, and the circulations in corresponding circuits, are the same in the two motions. The latter statement shews that vortices transform into vor- tices of equal strengths. It follows at once from Art. 145 that in the case of a closed simply-connected surface the algebraic sum of the strengths of all the vortices present is zero. Let us apply this to motion in a spherical stratum. The simplest case is that of a pair of isolated vortices situate at antipodal points ; the stream-Unes are then parallel small circles, the velocity varying inversely as the radius of the circle. For a vortex-pair situate at any two points A, B, the stream- lines are coaxal circles as in Art. 80. It is easily found by the method of stereographic projection that the velocity at any point P is the resultant of two velocities ic/27ra . cot ^di and ic/27ra . cot |02> perpendicular respectively to the great-circle arcs AP^ BP, where d^, d^ denote the lengths of these arcs, a the radius of the sphere, and ± k the strengths of the vortices. The centre f (see Art. 164) of either vortex moves perpendicular to AB with a velocity Kl2jra . cot ^AB, The two vortices therefore describe parallel and equal small circles, remaining at a constant distance from each other. Circular Vortices, 161. Let us next take the case where all the vortices present in the liquid (supposed unlimited as before) are circular, having the axis of 2; as a common axis. Let m denote the distance of any point P from this axis, v the * For further researches in this connection see Hill, **0n the Motion of Fluid part of which is moving rotationally and part irrotationally/' PkiL Trans. 1884; Love, "On the Stability of certain Vortex Motions," Proc Lond. Math, 80c, t. zxv. p. 18 (1893). t To prevent possible misconception it may be remarked that the centres of corresponding Tortices are not necessarily corresponding points. The paths of these centres are therefore not in general projective. 16—2 228 Vortex Motion [chap, vti velocity in the direction of w, and co the resultant vorticity at P. It is evident that u, v, oi are functions of Xy w only. Under these circumstances there exists a stream-function iff, defined as in Art. 94, viz. we have , ^??^_^=i f^V _L.^^].^\ . (2) " dx dm m \dx^ dm^ w dm) It is easily seen from the expressions (7) of Art. 148 that the vector {Fy Oy H) will under the present conditions be everjrwhere perpendicular to the axis of x and the radius m. If we denote its magnitude by S, the flux through the circle {Xy w) will be 27TmSy whence ilf=-mS (3) To find the value of ^ at (x, m) due to a single vortex-filament of cir- culation Ky whose co-ordinates are x\ to', we note that that element which makes an angle d with the direction of S may be denoted by w'S6, and there- fore by Art. 149 (1) . « Ktmn' [^ COB d jj^ ,.. ^ = "^^=-1^Jo — ^' (*) where r = {(« - a?')* + ©« + to'* - 2ojot' cos ^*. (5) If we denote by fj, r, the least and greatest distances, respectively, of the point P from the vortex, viz. f i« = (a - xy + (id ~ ©')^ r,* = (» - xy + (m + to')*, ... (6) we have r « = r i^ cos* Jfl + r,* sin« J0, 4toro' cos 6^ r^ -f- r ,« - 2r«, . . (7) and therefore ^ Stt L^*"^ "^ * ^ j V(ri« co8« ^e + V sii* P) - 2 / ' V(ri« cos« ^e + r,« sin« ^0)4$] (8) -'0 J The integrals are of the types met with in the theory of the 'arithmetico- geometrical mean.'* In the ordinary, less symmetrical, notation of 'com- plete' elliptic integrals we have *--£(«»■)'{(?-*)*•,(*)-?«,(»)} (9) prodded i. . , _ i; . j__^*^__,^. „o) The value of ^ at any assigned point can therefore be computed with the help of Legendre's tables. * See Gaylev, EUipitc Functions, Cambridge, 1876, c. ziii. 161] Circular' Vortices 229 A neater expression majt be obtained' by means of fLanden's trans- formation'*; thus ^«-£(^i+T.){Ji(A)~£i(A)}, (11) provided A = ^— * (12) ^j + ^1 To verify this, let AB be a straight line divided at P into two segments PA, PB of lengths 7*1 , r,, respectively ; and describe the circle on AB as diameter. C being the centre, and Q any point on the circumference, let the angles QCA, QPA be denoted by B, S, respec- tively ; and draw CN perpendicular to QP. If PQ=r, we have r»=ri«co82i^+r,«sin*i^, rdS=CQM , cos CQN=QNM (13) Also CP.'^^^-oosCQN^^^fS. r r ^ r CQ and therefore ^ co8^_ 6S PQds ^^ ds Qms pms f \q Hence I \/^^ since j PNdS^CP JQO3SdS=0. Now QN = ^{CQ^ - CP» sin« 5) = J (fi +rj) V(l - X' sin« 5), (16) « ^=?C=rT^; .........(16) The formula (14) may therefore be written i(r.-r,)f'?^^d^=2{^i(X)-^i(X)} (17) Jo r which brings (4) into the required form (11). The forms of the stream-lines corresponding to equidistant values of ^ are shewn on the next page. They are traced by a method devised by Maxwell, to whom the formula (11) is also duef. Expressions for the velocity-potential and the stream-function can also be obtained in the form of definite integrals involving Bessel's Fimctions. Thus, supposing the vortex to occupy the position of the circle ic » 0, to = a, it is evident that the portions of the positive side of the plane x^O which lie within and without this circle constitute two distinct equipotential surfaces. Hence, assuming that we have ^ = ^/c for x » 0, to < a, and ^ = for a; » 0, to > a, we obtain from Art. 102 (2) <t>^\Ka\ e-^'Jo{kw)J^{ka)dk, (18) J * See Cayley, 2.c. t EledricUy and Jiagneiism, Arto. 704, 705. See also Minchin, PhiL Mag. (6), t. zxx7. (1803); Nagaoka, PhO. Mag. (6), t. vi (1903). 230 Vortex Motion and t^eiefoce, in accordance with Art. 100 (5), = - i Kom I e-*'Ji {km) J, (ka) dk. These formulae relate of course to the region x > 0*. [chap, vn (19) > The [onnnlk for ^ See alao Nkgaoka, Le. 161-162] StreamrlAnes of a Vortex-Ring 231 It was shewn in Art. 150 that the Value of ^ is that due to a system of double sources distributed with uniform density k over the interior of the circle. The values of <f> and tft for a uniform distribution of simple sources over the same area have been given in Art. 102 (11). The above formulae (18) and (19) can thence be derived by differentiating with respect to x, and adjusting the constant factor*. 162: The energy of any system of circular vortices having the axis of x as a common axis, is = — Trp / j^Kodxdw = — npYiKift, . • (1) by a partial integration, the integrated terms vanishing at the limits. We have here used ic to denote the strength coSxStn of an elementary vortex- filament. Again the formula (7) of Art. 153 becomes f T =*= iirp //(row — xv) XDcodxdy = 2mpl!iKXD {mu — xv) (2) The impulse of the system obviously reduces to a force along Ox. By Art. 152 (6), -P ~ ip lliyt — ^) dxdydz = irp ijwhadxdm *= TrpSicw* (3) If we introduce the symbols Xq, Wq defined by the equations .these determine a circle whose position evidently depends on the strengths and the configuration of the vortices, and not on the position of the origin on the axis of symmetry. It may be called the 'circular axis' of the whole system of vortex-rings. Since k is constant for each vortex, the constancy of the impulse shews, by (3) and (4), that the circular axis remains constant in radius. To find its motion parallel to x, we have, from (4), Sic . ©0* . -j^ = S/ctD* ^ + 2Eicrox -^ = Ektd {mu 4- 2xu) (5) * Other ezpreadons for and ^ can he obtained in terms of zonal spherical harmonics. ThoB the value of is given in Thomson and Tait, Art. 546; and that of ^ can be deduced by the formulae (11), (12) of Art. 95 ante. The elliptic -integral forms are however the most useful for purposes of interpretation. t At any point in the plane z=0 we have y=iff, (=0, 17=0, i;=l<a,v==v; the rest follows by symmetry* 232 Vortex Motion [chap, vn With the help of (2) this can be ptit in the fonn Sic. Too* . ^® = ^ + 32ic (a; -Zo)mv,.,. (6) where the added term vanishes, since JIkwv == on account of the constancy of the mean radius (mo). 163. Let us now consider, in particular, the case of an isolated vortex- ring the dimensions of whose cross-section are small compared with th^ radius (tOo)* It has been shewn that * - - nil'- Citt) - *. CTrf;)} '" + '.) '»''^*^. ■ • m where r^, rj are defined by Art. 161 (6). For points («, in) in or near the substance of the vortex, the ratio r i/fs is small, and the modulus (A) of the elliptic integrals is accordingly nearly equal to. unity. We then have yi(A) = iIog^„ ^i(A)=l (2) approximately*, where A' denotes the complementary modulus, viz. A'2 = 1 - A2 = j-^^^^ ,- , (3) or A'* = ^Ti/fj, nearly. Hence at points within the substance of the vortex the value of ^ is of the order ict^o log (tOo/c), where e is a small linear magnitude comparable with the dimensions of the section. The velocities at such points, depending (Art. 94) on the differential coefficients of ^, will be of the order ic/c. We can now estimate the magnitude of the velocity dxQ/dt of translation of the vortex-ring. By Art. 162 (1), T is of the order pk^Dq log (otq/c), and v is, as we have seen, of the order k/c ; whilst x — a;© is of course of the order €. Hence the second term on the right-hand side of the formula (6) of the preceding Art. is, in the present case, small compared with the first, and the velocity of translation of the ring is of the order k/wq . log (mo/^)> ^^^ approximately constant. An isolated vortex-ring moves then, without sensible change of size, parallel to its rectilinear axis with nearly constant velocity. This velocity is smaU compared with that of the fluid in the immediate neighbourhood of the circular axis, but may be greater or less than ^k/wq, the velocity of the fluid at the centre of the ring, with which it agrees in direction. For the case of a circiUar section more definite results can be obtained as follows. If we neglect the variations of v and a> over the section, the formulae (1) and (2) give * See Cayley, Elliptic FuncHonn, Arts. 72, 77 ; and Maxwell, Lc, 162^164] ; Speed of a Vortex-Ring 233 QTy H w6 intrdduoe polar co-ordinatee {s, x) ^ the plAne of the section, where a is the radius of the section. Now f^ log Mx' = f^ log {^ + *-* - 28^" cos ix - x')}* dx\ and this definite integral is known to be equal to 2n log «^ or 2ir log «, according as 9' ^ «. Hence, for points within the section, ^= -»79i J' (lOg?^0 .2) ^^r _^^^j- ^i^Sjp _2).W = .Woa«{log?j2-f-i^.} (6) The only variable part of this is the term laxff^; this shews that to our order of approanmation the stream-lines within the section are concentric circles, the velocity at a distance 8 from the centre being i»8. Substituting in Art. 162 ( 1 ) we find The last term in Art. 162 (6) is equivalent to In our present notation, where k denotes the strength of the whole vortex, this is equal to iK^vrJn, Hence the formula for the velocity of translation of the vortex becomes* dxn t=i^.K-?-i} <'> The vortex-ring carries with it a certain body of irrotationaUy moving fluid in its career; cf. Art. 166, 2°. According to the formula (7) the velocity of translation of the' vortex will be equal to the velocity of the fluid at its centre when wJa=SQf about. The accompanying mass will be ring-shaped or not, according as ^sja exceeds or falls short of this critical value. The ratio of the fluid velocity at the periphery of the vortex to the velocity at the centre of the ring is 2aamjK, or wjira. For a =1^9^11 * this \r equal to 32, about. 164. If we have any number of circular vortex-rings, coaxal or not, the motion of any one of these may be conceived as made up of two parts, one due to the ring itself, the other due to the influence of the remaining rings. The preceding considerations shew that the second part is insignificant compared with the first, except when two or more rings approach within a very small distance of one another. Hence each ring will move, without * This result was given without proof by Sir W. Thomson in an appendix to a translation of Hehnholtz' paper, PhU. Mag. (4), t. xxxiii. p. 611 (1867) [Paper9, t. iv. p. 67]. It was verified by Hicks, PhiL Trans. A, t. clxxvi. p. 756 (1886); see also Gray, "Notes on Hydrodynamics," PhiL Mag, (6), t. xxviii p. 13 (1914). 234 Vortex Motion [pHAP. vn sensible change of shape or size, with nearly nniform velocity in the' direction of its rectilinear axis, until it passes within a short distance of a second ring.' A general notion of the result of the encounter of two rings may, in particular cases, be gathered from the result given in Art. 149 (3). Thus, let us suppose that we have two circular vortices having the same rectilinear axis. If the sense of the rotation be the same for both, the two rings will advance, on the whole, in the same direction. One effect of their mutual influence will be to increase the radius of the one in front, and to contract the radius of the one in the rear. If the radius of the one in front become larger than that of the one in the rear, the motion of the former ring will be retarded, and that of the latter accelerated. Hence if the conditions as to relative size and strength of the two rings be favourable, it may happen that the second ring will overtake and pass through the first. The parts played by the two rings will then be reversed ; the one which is now in the rear will in turn overtake and pass through the other, and so on, the rings alternately passing one through the other*. If the rotations be opposite, and such that the rings approach one another, the mutual influence will be to enlarge the radius of each. If the two rings be moreover equal in size and strength, the velocity of approach will continually- diminish. In this case the motion at all points of the plane which is parallel to the two rings, and half-way between them, is tangential to this plane. We may therefore, if we please, regard the plane as a fixed" boundary to the fluid on either side, and so obtain the case of a single vortex-ring moving directly towards a fixed rigid wall. The foregoing remarks are taken from Helmholtz' paper. He adds, in conclusion, that the mutual influence of vortex-rings may easily be studied experimentally in the case of the (roughly) semicircular rings produced by drawing rapidly the point of a spoon for a short space through the surface of a liquid, the spots where the vortex-filaments meet the surface being marked by dimples. (Cf. Art. 27.) The method of experimental illustration by means of smoke-rings f is too well-known to need description here. A beauti- ful variation of the experiment consists in forming the rings in water, the substance of the vortices being coloured J. The motion of a vortex-ring in a fluid limited (whether internally or externally) by a fixed spherical surface, in the case where the rectilinear axis of the ring passes through * The corresponding case in two dimensions was worked out and illustrated graphically by Grobli, Ix. ante p. 218 ; see also Love, " On the Motion of Paired Vortices with a Common Axis,'* Proc, Lond. Maih. 8oc. t. zxv. p. 185 (1894). t Kousch. **Ueber Ringbildung der Fliissigkeiten/* Pogg. Ann. t. ex. (1800); Tait, Beeent Advances in Phyaical Science, London, 1876, c. xii. X Reynolds, **0n the Resistance encountered by Vortex Rings &c.,*' Brit, Ass, Sep, 1876; Nature, t xiv. p. 477. 164-165] Mutual Influence of Vortex-Rings 235 the centre of the sphere, has been investigated by Lewis*, by the method of * images.* The following simplified proof is due to Larmor f. The vortex-ring is equivalent (Art. 150) to a spherical sheet of double-sources of uniform density, concentric with the fixed sphere. The * image' of this sheet will, by Art. 96, be another uniform concentric double-sheet, which is, again, equivalent to a vortex-ring coaxal with the fijrst. It easily follows from the Art. last cited that the strengths (jc, k') and the radii (or, w') of the vortex-ring and its image are connected by the relation KOT* + ic'w'* =0. (1) The argument obviously appties to the case of a re-entrant vortex of any form, provided it he on a sphere concentric with the boundary. On the Conditions for Steely Motion. 165. In steady motion, i,e, when dt ' dt "' dt "' the equations (2) of Art. 6 may be written du , dv , dw , ^ . dCl Idp „. Hence, if as in Art. 146 we put /=/^ + h»+a (2) we have ^ = ^^ "" ^» ^ "" ^^ "" *^^' ^ "^ ^'^ " ^^ (^) It f oUows that w 1^' + t; ^ + tr 1^' - 0, ox oy oz ^ dx^'^dy ^^ dz "' so that each of the surfaces x' == const, contains both stream-lines and vortex-lines. If further 8n denote an element of the normal at any point of such a surface, we have ^-goisinjS, (4) where q is the current- velocity, w the vorticity, and j8 the angle between the stream-line and the vortez-Une at that point. Hence the conditions that a given state of motion of a fluid may be a possible state of steady motion are as follows. It must be possible to draw in the fluid an inflnite system of surfaces each of which is covered by * "On the Images of Vortices in a Spherical Vessel,*' Quart. Jaum, Math. t. xvl p. 338 (1879). t '* Electro-magnetic and other Images in Spheres and Planes," QttarL Joum, Math. t. xxiiL p. 94 (1889). 236 Vortex Motion [chap, vn a network of stream-lineg and vortex-lines, and the product jco sin j3 Sn must be constant over each such stirface, Sn denoting the length of the normal drawn to a consecutive surface of the system*. These conditions may also be deduced from the considerations that the stream-lines are, in steady motion, the actual paths of the particles, that the product of the angular velocity into the cross-section is the same at all points of a vortex, and that this product is, for the same vortex, constant with regard to the time. The theorem that the function x'» defined by (2), is constant over each surface of the above kind is an extension of that of Art. 21, where it was shewn that x is constant along a stream-line. The above conditions are satisfied identically in all cases of irrotational motion, provided of course the boundary-conditions be such as are consistent with the steady motion. In the motion of a Uquid in two dimensions (xy) the product qBn is constant along a stream-line; the conditions in question then reduce to this, that the vorticity ^ must be constant along each stream-line, or, by Art. 59 (5), . g+p-/w P) where f (iff) is an arbitrary function of ^f- This condition is satisfied in all cases of motion in concentric circles about the origin. Another obvious solution of (5) is Vr =i (Ax* +2Bxy +Cy^) (6) in which case the stream-lines are similar and coaxal conic?. The angular velocity at any point i3^(A-¥ C), suid is therefore uniform. Again, if we put/ (tfr) = - k^, where A; is a constant, and transform to polar co-ordinates r, 6, we get di^^rW'-^de^^^^-^ ^^^ which is satisfied (Art. 101) by ^=CJ,{hr) ^!^Ud (8) This gives various solutions consistent with. a fixed circular boundary of radius a, the admissible values of h being determined by jr,(ifca)=0. :....(9) * See a paper *'0n the Conditions for Steady Motion of a Fluid/' Proc, Lond, Math. 8oc, t. ix. p. 91 (1878). t Cf. Lagrange, Nouv. Mim, de FAead. de Berlin, 1781 [Oeuvres, t. iv. p. 720]; and Stokes, ''On the Steady Motion of Inoompressible Fluids," Camb, Trans, t. vii (1842) [Papets, t. L p. 15]. 165] Conditions for Steady Motion 237 Suppose, for example, that in an unlimited ma38 of fluid the stream-function is ^ =CJi (At) sin ^, (10) within the circle r=a, whilst outside this circle we have ylr = u(r-'^ame (11) These two values of tfr agree for r=a, provided Jj (ka) = 0. Moreover, the tangential velocity at this circle will be continuous, provided the two values of 8^/dr are equal, »'.€. if If we now impress on everything a velocity U parallel to Ox, we get a species of cylindrical vortex travelling with velocity U through a liquid which is at rest at infinity. The smallest of the possible values of ik is given by ka/v = 1-2197; the relative stream-lines inside the vortex are then given by the lower diagram on p. 280, provided the doUed circle be taken as the boundary (r=a). It is easily proved, by Art. 157 (1), that the * impulse* of the vortex is represented by 2jrpa*U, In the case of motion syinmetrical about an axis (x), we have q . 27rwi8n constant along a stream-line, m denoting as in Art. 94 the distance of any point from the axis of symmetry. The condition for steady motion then is that the ratio ai/t? must be constant along any stream-line. Hence, if ^ be the stream-function, we must have, by Art. 161 (2), ^^&.-i¥.-'"m (.3) dx^ 3c5* w dm where /(0) denotes an arbitrary function of ^*. An interesting example is furnished by Hill*s * Spherical Vortex f.' If we assume Vr =i^ar« (a^-r*), (14) where r* =a:* +m\ for all points within the sphere r=a, the formula (2) of Art. 161 makes so that the condition of steady motion is satisfied. Again it is evident, on reference to Arts. 96, 97, that the irrotational flow of a stream with the general velocity - U parallel to the axis, past a fixed spherical surface r =a, is given by Vr=JC7or«(l-^ (16) The two values of y^r agree when r=a; this makes the normal velocity zero on both sides. In order that the tangential velocity may be continuous, the values of d^jt/ar must also agree. Remembering that or =r sin ^, this gives A = -^U/a^ and therefore a> =^Uw/a* (16) The sum of the strengths of the vortex-filaments composing the spherical vortex is BUa, * This result is due to Stokos, 2.c. t '*0n a Spherical Vortex," Phil Trans. A, t. clxxxv. (1894). 238 Vortex Motion [chap, vn The figure shews the stream-lines, both inside and outside the vortex ; they are drawn, as usual, for equidistant values of ^. If we impress on everything a velocity U parallel to x, we get a spherical vortex advancing with constant velocity U through a liquid which is at rest at infinity. By the formulae of Art. 162, we readily find that the square of the * mean-radius ' of the vortex is fa*, the 'impulse' is 2vpa^U, and the energy is ^{^pa*UK As explained in Art. 146, it is quite unnecessary to calculate formulae for the pressure, in order to assure ourselves that this is* continuous at the surface of the vortex. The con- tinuity of the pressure is already secured by the continuity of the velocity, and the constancy of the circulation in any moving circuit. 166. As already stated, the theory of vortex motion was originated by Helmholtz in 1858. It acquired additional interest when, in 1867, Kelvin suggested* the theory of vortex atoms. As a physical theory, this lies outside our province, but it has given rise to a great number of interesting investigations, to which some reference should be made. We may mention the investigations as to the stability and the periods of vibration of recti- linear f and annular]: vortices; the similar investigations relating to hollow vortices (where the rotationally moving core is replaced by a vacuum §) ; and the calculations of the forms of boundary of a hollow vortex which are con- sistent with steady motion ||. A summary of some of the leading results has been given by Love^f. * Lc, ante p. 216. t Sir W. Thomson, Ic, ante p. 224. X J. J. Thomaon, l.c. ante p. 209; Dyson, Phil. Trans. A, t. clxxxiv. p. 1041 (1893). § Sir W. Thomson, l.c.; Hicks, "On the Steady Motion and the Small Vibrations of a Hollow Vortex," PhU. Trans. 1884; Pocklington, "The Complete System of the Periods of a Hollow Vortex Ring," Phil Trans. A, t. clxxxvi p. 603 (1896); Carslaw, "The Fluted Vibrations of a Circular Vortex-Ring with a HoUow Core," Proc. Land. Math. Soc. t. xxviii. p. 97 (1896). ii Hicks, l.c. ; Pocklington, " Hollow Straight Vortices," Camb. Proc. t. viiL p. 178 (1894). II l.c. anUp. 183. 16&-167] HilTs Spherical Vortex 239 ClebscVs Transformation. 167. Another matter of some interest, which can however only be briefly touched upon, is Clebsch's transformation of the hydrodynamical equations*. It is easily seen that the component velocities at any one instant can be expressed in the forms where <^, X, la are functions of z, y, z, provided the component rotations can be put in the forms 8(X^) d{\,^) 8(X,/x) ^~d(y.zy '^''d(z,xy ^-d(x,y) ^^^ Now if the differential equations of the vortex -lines, viz. dx _dy _dz y"7"T ^^^ be supposed integrated in the form a =const., /3 =const (4) where a, /3 are functions of x, y, z, we must have ^ p 8(a,g ) _p 8(a, ff) ^_p 8(a, ff) ^-^8(y,z)' ''-^8(«,a;)' '■~^8(a:,y) ^^^ where P is some function of x, y, zf* Substituting these expressions in the identity oa? cy 02 wefind 8 (P, a, ^) ^Q 8 (a?, y, 2) ' ' • which shews that P is of the form / (a, /3). If X, /ui be any two functions of a, /3, we have 3(y.*) S(a,^) 8(y.z)' » ' » • and the eqtiations (6) will therefore reduce to the form (2), provided X, m be choeen so that lgrg=/(-^) (') which can obviously be satisfied in an infinity of ways. It is evident from (2) that the intersections of the surfaces X = const., /i = const, are the vortex-lines. This suggests that the functions X, /jl which occur in (1) may be supposed to vary continuously with t in such a way that the surfaces in question move with the fluid $. Various analytical proofis of the possibility of this have been given ; the simplest, perhaps, is by means of the equations (2) of Art. 16, which give (as in Art. 17) udx +vdy+wdz=UQda +VQdb +WQdc -dx (8) It has been proved that we may assume, initially, ti^da+VQdb+WQdc^ -(2</>o+X(2fu (9) * "Ueber eine aUgemeine Transformation d. hydrodynamlBchen Qleichungen," Crelle, t. liv. (1857) and t. IvL (1860). See also Hill, QtuirL Joum, Math, t. xvii. (1881), and Camb. Trans. t. xiv. (1883). t GL Forsyth, DifferenJtidl Egnations, Art. 174. X It must not be overlooked that on account of the insuflGioient detemunacy of \, fi these functions may vary continuously with t without relating always to the same particles of fluid. 240 Vortex Motion [chap, vn Hence, considering space- variations at time t^ we shall have ■ * udx .+vdy +wdz = -d^i +Xrff*, (10) f . . . .' where^ ^ s^, +x> aad X, fi have the same values as in (9), but are now expressed in terms of X, y, Zy t. Since, in the 'Lagrangian* method the independent space- variables relate to the individual particles, this proves the theorem. On this understanding the equations of motion can be integrated, provided the extraneous forces have a potential, and that p is a function of p only. We have ^w ft J. « 8tt / 5X d\ dK\ 9u / du. du 3u\ 9X ~dx\~dt'^''dt)'^ Dldx~ Dtdx' ^"' and therefore, on the present Assumption that D\/Dt =0, Dn/Dl =0, J^^.1^.0=|-X| (12) by Art. 146 (5), (6). An arbitrary function of t is here supposed incorporated in d^/8^ If the above condition be not imposed on X, /x, we have, writing ^=/?-i^-°-t-4 (13) Dtdx''lH^~'dx' Dtdy''Dtdy~ dy* Dt dz Dt dz" Zz'"'^^ Hence Tr^^=^ d^) shewing that H is of the form / (X, ^, t) ; and DK_ m Dfi_dH IH~~d,i' Dt~dX (^"' CHAPTER VIII TIDAL WAVES 168. One of the most interesting and successful applications of hydro- dynamical theory is to the small oscillations, under gravity, of a liquid having a free surface. In certain cases, which are somewhat special as regards the theory, but very important from a practical point of view, these oscillations may combine to form progressive waves travelling with (to a first approxi- mation) no change of form over the surface. The term ^ tidal,' as applied to waves, has been used in various senses, but it seems most natural to confine it to gravitational oscillations possessing the characteristic feature of the oceanic tides produced by the action of the sun and moon. We have therefore ventured to place it at the head of this Chapter, as descriptive of waves in which the motion of the fluid is mainly horizontal, and therefore (as will appear) sensibly the same for all particles in a vertical line. This latter circumstance greatly simplifies the theory. It will be convenient to recapitulate, in the first place, some points in the general theory of small oscillations which will receive constant exemplification in the investigations which follow*. The theory has reference in the first instance to a system of finite freedom, but the results, when properly inter- preted, hold good without this restriction f. Let ?i, ?2, . . . Jn be n generalized co-ordinates serving to specify the con- figuration of a dynamical system, and let them be so chosen as to vanish in the configuration of equilibrium. The kinetic energy T will be a homogeneous ^ quadratic function of the generalized velocities qi, q^f > • » ^m ^7 2T = aiiji* + «2292* + • . • + 2aijj^ig2 + • • • > (1) where the coefficients are in general functions of the co-ordinates 9i, 929 • - • 9n > but may in the application to small motions be supposed constant, and to have the values corresponding to 9x> ?29 • • • 9n "= 0. Again, if (as we shall * For a fuller account of the general theory see Thomson snd Tait, Arts. 337, . . . ; Rayleigh, Tlieory of Sound, c. iv. ; Routh, Elementary Rigid Dynamics (6th ed.), London, 1807» c. ix. ; Whittaker, Analytical Dynamics, o. vii t The steps by which a rigorous transition can be made to the case of infinite freedom have been investigated by Hilbert, Q&L Nachr. 1004, p. 40. L. H. 16 242 Tidal Waves [chap, vin suppose) the system is 'conservative/ the potential energy F of a small displacement is a homogeneous quadratic function of the component displacements q^^ 9if -- - 9n* ^^^ X^^ the same understanding) constant coefficients, say 2V = CuJx* H- Cjajj* + . . . + 2ci,grigr, + (2) By a real* linear transformation of the co-ordinates 9i, 92, . . . 9n ^^ ^ possible to reduce T and V simultaneously to sums of squares; the new variables thus introduced are called the * normal co-ordinates' of the system. In terms of these we have 2r = aigi« + a,g,* + ... H-a„g««, (3) 2F = CiJi« + c,ft« + ... -hCnqn^ (4) The coefficients a^y a^, . » . a^ are called the ^principal coefficients of inertia' ; they are necessarily positive. The coefficients Ci, c^^ . , . c^ may be called the ' principal coefficients of stability ' ; they are all positive when the undisturbed configuration is stable. When given extraneous forces act on the system, the work done by these during an arbitrary infinitesimal displacement A^^, Aq^, . . . Ag^ may be expressed in the form GiA?x + QaAj,+ ... +Q„A?„. (5) The coefficients Qi, Q^, ... Qn a*re then called the 'normal components of disturbing force.' In the appUcation to infinitely small motions Lagrange's equations take the form «1.4'l + «2r?2 + • • • -^^Itqi + C2f?« + ...=* Or (7) or, in the case of normal co-ordinates, «r9r + Crqr = Qr (8) •It is easily seen from this that the dynamical characteristics of the normal co-ordinates are (F) that an impulse of any normal type produces an initial motion of that type only, and (2°) that a steady disturbing force of any type maintains a displacement of that type only. To obtain the free motions of the system we put Qr = 0. Solving (8), we find qr = Ar COS (art + €r\ (9) where a '-©• ™ * The algebraic proof of this involves the assnmption that one at least of the functions T, V is essentially positive. In the present case T of conrse fulfils this condition. 168] Small Oscillations 243 and A^t e^ are arbitrary constants*. Hence a mode of free motion is possible in which any normal co*ordinate q^ varies alone, and the motion of any particle of the system, since it depends linearly on q^t will be simple-harmonic, of period 2^/(7^; moreover the particles will pass simultaneously through their equilibrium positions. The several modes of this character are called the 'normal modes' of vibration of the system; their number is equal to that of the degrees of freedom, and any free motion whatever of the system may be obtained from them by superposition, with a proper choice of the * amplitudes' (-4^) and * epochs ' (c^). In certain cases, viz. when two or more of the free periods (2w/a) of the system are equal, the normal co-ordinates are to a certain extent indeterminate, i,e, they can be chosen in an infinite number of ways. An instance of this is the spherical pendulum. Other examples will present themselves later ; see Arts. 191, 200. If two (or more) normal modes have the same period, then by compounding them, with arbitrary amplitudes and epochs, we obtain a small oscillation in which the motion of each particle is the resultant of simple-harmonic vibrations in different directions, and is therefore, in general, elliptic-harmonic, with the same period. This is exemplified in the spherical pendulum; an important instance in our own subject is that of progressive waves in deep water (Chapter ix.). If any of the coefficients of stability (c^) be negative, the value of a^ is a pure imaginary. The circular function in (9) is then replaced by real ex- ponentials, and an arbitrary displacement will in general increase until the assumptions on which the approximate equation (8) is based become untenable. The undisturbed configuration is then reckoned as unstable. The necessary and sufficient condition of stability (in the present sense) b that the potential energy V should be a minimum in the configuration of equilibrium. To find the effect of disturbing forces, it is sufficient to consider the case where Q^ varies as a simple-harmonic function of the time, say Q^ = 0^ cos (a« -f €), (11) where the value of a is now prescribed. Not only is this the most interesting case in itself, but we know from Fourier's Theorem that, whatever the law of variation of Q^ with the time, it can be expressed by a series of terms such as (11). A particular integral of (8) is then g^° r ^'^a <^Qs(<^ + ^) (12) * The ratio <r/2T measures the 'frequency' of the oscillation. It is convenient to have a name for the quantity a itself; the term * speed* has been used in this sense by Kelvin and 0. H. Darwin in their researches on the Tides. 16—2 244 Tidal Waves [chap, vra This represents the 'forced oscillation' due to the periodic force Q^. In it the motion of every particle is simple-harmonic, of the prescribed period 2^/<7, and the extreme displacements coincide in time with the maxima and minima of the force. A constant force equal to the instantaneous value of the actual force (11) would maintain a displacement n 5^ « ^ cos (a< + €), (13) the same, of course, as if the inertia-coefficient a^ were null. Hence (12) may be written where g^ has the value (10). This very useful formula enables us to write down the effect of a periodic force when we know that of a steady force of the same type. It is to be noticed that q^ and Q^ have the same or opposite phases according as a $ a^, that is, according as the period of the disturbing force is greater or less than the free period. A simple example of this is furnished, by a simple pendulum acted on by a periodic horizontal force. Other important illustrations will present themselves in the theory of the tides*. When a is very great in comparison with a,., the formula (12) becomes Q ?r = - -2^ cos (<rf H- €) ; (15) the displacement is now always in the opposite phase to the force, and depends only on the inertia of the system. If the period of the impressed force be nearly equal to that of the normal mode of order r, the amplitude of the forced oscillation, as given by (14), is very great compared with g^. In the case of exact equality, the solution (12) fails, and must be replaced by gr^ = 5e sin (a« + e), (16) where, as is verified immediately on substitution, B = C,./2<7a^. This gives an oscillation of continually increasing amplitude, and can therefore only be accepted as a representation of the initial stages of the disturbance. Another very important property of the normal modes may be noticed. If by the introduction of constraints the system be compelled to oscillate in any other prescribed manner, the configuration at any instant c€Ui be specified by one variable, which we will denote by 6, In terms of this we shall have qr=Br6, * Cf. T. Young, "A Theory of Tides," NichoUorCa Journal, t. xzzv. (1813) [MidceUaneow Works, London, 1864, t. u. p. 262]. 168] The(yry of Normal Modes 245 where the quantities Br are certain oonstants. This makes 2T=(Bi«ai +B,«a, + . . . +B»«an)^, (17) 2F=(Bi«Ci-fB,*c,+ ... +B»«0^ (18) If ^ a cos (crt +€), the constancy of the energy {T + V) requires Hence o-' is intermediate in value between the greatest and least of the quantities Crja^ ; in other words, the frequency of the constrained oscillation is intermediate between the greatest and least frequencies corresponding to the normal modes of the system. In par- ticular, when a system is modified by the introduction of a constraint, the frequency of the slowest natural oscillation is irkcreased. Moreover, if the constrained type differ but slightly from a normal type (r), cr' will differ from (v/Or by a small quantity of ike second order. This gives a valuable method of estimating approximately the frequency in cases where the normal types cannot be accurately determined*. It may further be shewn that in the case of a partial constraint, which merely reduces the degree of freedom from n to n - 1, the periods of the modified system separate those of the original onef . It is of some interest in the present connection to recall a remark made by Lagrange in the M&aniqiu AnalyiiqueX to the effect that if in the equations of type (7), where the co-ordinates are not assumed to be normal, we put Qr=0, and assume qr^Ar^\ ...(20) the resulting equations are identical with those which determine the stationary values ( - X*) of the expression ^1-^1 "^ ^-^1 + " » + 2ci2AiA^ + . . . ^ .gj. ^1^1*+ ^^2*+ •'• + ^Oiji^i^, + Since T is essentially positive the denominator cannot vanish, and the expression has therefore a minimum value. It is moreover possible, starting from this property, to construct a proof that the n values of X' are all real§. They are obviously all negative if F be essentially positive. Rayleigh*s theorem is also closely related to the Hamiltonian formula (19) of Art. 135, as we may see by assuming qr = ArBUKrt (22) and taking ^^ = 0, ^ = 2ir/<r. The modifications which are introduced into the theory of small oscillations by the consideration of viscous forces will be noticed in Chapter xi. * Rayleigh, "Some General Theorems relating to Vibrations," Proc, Lond, Math. 8oc. t. iv. p. 367 (1874) [Paperd, t. L p. 170], and Theory of Sound, o. iv. The method is elaborated by Bitz, Joum.f4kr Math., t. cxxxv. p. 1 (1908), and Ann. der Physik, t. xxviil (1909) [Gesammelte Werke, Paris, 1911, pp. 192, 2S6]. t Routh, Elementary Rigid Dynamics, Art. 67 ,- Rayleigh, Theory of Sound (2nd ed.). Art. 92 a ; Wbittaker, Analytical Dynamics, Art. 81. X Oeuvres, t. xi, p. 380. § See Poinoar^, Jowm. de Math. (6), t. ii. p. 83 (1896). 246 Tidal Waves [chap, vm Long Waves in Canals. 169. Proceeding now to the special problem of this Chapter, let us begin with the case of waves travelling along a straight canal, with horizontal bed, and parallel vertical sides. Let the axis of x be parallel to the length of the canal, that of y vertical and upwards, and let us suppose that the motion takes place in these two dimensions x, y. Let the ordinate of the free surf ace, corresponding to the abscissa x, at time f, be denoted by y^ + % where yo ^ the ordinate in the undisturbed state. As already indicated, we shall assume in all the investigations of this Chapter that the vertical acceleration of the fluid particles may be neglected, or, more precisely, that the pressure at any point (a;, y) is sensibly equal to the statical pressure due to the depth below the free surface, viz. ' . p-Po = 9p(yo + v-y)y (1) where p^ is the (uniform) external pressure. H«°c« l=^''i (2) This is independent of y, so that the horizontal acceleration is the same for all particles in a plane perpendicular to x. It follows that all particles which once lie in such a plane always do so ; in other words, the horizontal velocity uiB a, function of x and t only. The equation of horizontal motion, viz. du du ^ 1 dp dt dx" pdx' is further simplified in the case of infinitely small motions by the omission of the term udu/dxy which is of the second order, so that Now let f = Jw(ft; i.e. ^ is the time-integral of the displacement past the plane x, up to the time t In the case of smaU motions this will, to the first order of small quantities, be equal to the displacement of the particles which originally occupied that plane, or again to that of the particles which actually occupy it at time t. The equation (3) may now be written ar' = -^g w 169] Waves in Uniform Canal 247 The equation of contmuity may be found by calculating the volome of fluid which has, up to time ty entered the space bounded by the planes x and X + Sx; thus, if A be the depth and b the breadth of the canal, - g^ (f A6) Sx = -ribhx, ♦ The same result comes from the ordinary form of the equation of con- tinuity, viz. i+i=» <«) "-/Is^-^S <" if the origin be (for the moment) taken in the bottom of the canal. This formula is of interest as shewing that the vertical velocity of any particle is simply proportional to its height above the bottom. At the free surface we have y = A + 17, v = 3iy/%, whence (neglecting a product of small quantities) di^^^d^t ^^^ From this (5) follows by integration with respect to L Eliminating 17 between (4) and (5), we obtain a? = ^*^« <^) The elimination of ^ gives an equation of the same form, viz. ^-^"^ « The above investigation can readily be extended to the case of a imiform canal of any form of section*. If the sectional area of the un- disturbed fluid be Sy and the breadth at the free surface 6, the equation of continuity is - 4 (^5) 8x = 7,68x, (11) whence ly = — A^, (12) as before, provided h =■ S/b, i.e. h now denotes the mean depth of the canaL The dynamical equation (4) is of course unaltered. * Kelland, Trans. JR. 8. Sdin. t. xiv. (1839). 248 Tidal Waves [chap, vm 170. The equation (9) is of a well-known type which occurs in several physical problems, e.g. the transverse vibrations of strings, and the motion of sound-waves in one dimension. To integrate it, let us write, for shortness, c^^gK (13) and a? — c^ = x^^ x •\- ct ^= x^. In terms of x^ and x^ as independent variables, the equation takes the form dx-i dx^ The complete solution is therefore ^ = F(x^ct) +f(x + ct), (14) where F, / are arbitrary functions. The corresponding values of the particle-velocity and of the surface- elevation are given by "^ I..- (15) 5=-F(a5-ce)-/(a? + c<).J The interpretation of these results is simple. Take first the motion represented by the first term in (14), alone. Since F {x-^ ct) is imaltered when t and x are increased by r and cr, respectively, it is plain that the dis- turbance which existed at the point x at time t has been transferred at time < + T to the point x -\- cr. Hence the disturbance advances unchanged with a constant velocity c in space. In other words we have a * progressive wave' travelling with constant velocity c in the direction of a-positive. In the same way the second term of (14) represents a progressive wave travelling with velocity c in the direction of x-negative. And it appears, since (14) is the complete solution of (9), that any motion whatever of the fluid, which is subject to the conditions laid down in the preceding Art., may be regarded as made up of waves of these two kinds. The velocity (c) of propagation is, by (13), that *due to' half the depth of the undisturbed fluid*. The foUowing table, giving in round numbers the velocity of wave-propagation for various depths, will be of interest later in connection with the theory of the tides. The last column gives the time a wave would take to travel over a distance equal to the earth*8 circumference (2«ra). In order that a 'long* wave should traverse this distance in 24 hours, the depth would have to be about 14 miles. It must be borne in mind that * Lagrange, Nouv. nUm, de VAcad, de Berlin, 1781 [OeuvreSf t. i p. 747]. 170-171] Wave- Velocity 249 these numerioal results are only applicable to waveis satisfying the conditions above postulated. The meaning of these conditions will be examined more particularly in Art 172. h c c 2ira/e (feet) (feet per sec.) (sea-miles per hour) (hours) 312i 100 60 360 1250 200 120 180 5000 400 240 90 11250* 600 360 60 20000 800 480 45 171. To trace the effect of an arbitrary initial disturbance, let us suppose that when ^ = we have ^=^(a;), l^'^ix) (16) The functions F'^f which occur in (15) are then given by F'{x) = -l{4>(x) + ^{x)},) fix)^ H<f> i<») - 'I' im ^ ' Hence if we draw the curvea y = i)i, y = rj^, where Vi = ih{^{x)+<f,{x)},\ rit = hf^{^l»(x)-4>(x)),\ ^""^ the form of the wave-profile at any subsequent instant t is found by displacing these curves parallel to x, through spaces ± ct, respectively, and adding (alge- braically) the ordinates. If, for example, the original disturbance be confined to a length I of the axis of a;, then after a time 2/2c it will have broken up into two progressive waves of length I, travelling in opposite directions. In the particular case where in the initial state f = 0, and therefore <l> {x) = 0, we have lyi = ^a 5 *t® elevation in each of the derived waves is then exactly half what it was, at corresponding points, in the original disturbance. It appears from (16) and (17) that if the initial disturbance be such that ^ = ± Tj/h . c, the motion will consist of a wave system travelling in one direction only, since one or other of the functions F' and/' is then zero. It is easy to trace the motion of a surface-particle as a progressive wave of either kind passes it. Suppose, for example, that i = F{x-ct), (19) and therefore f = c? (20) * This Jb probably comparable in order of magnitude with the mean depth of the ocean. 250 TidcU Waves [ohap. vm The particle is at rest until it is reached by the wave ; it then moves forward with a velocity proportional at each instant to the elevation above the mean level, the velocity being in fact less than the wave- velocity c, in the ratio of the surface-elevation to the depth of the water. The total displacement at any time is given by ^ ~ hi ^^^' This integral measures the volume, per unit breadth of the canal, of the portion of the wave which has up to the instant in question passed the particle. Finally, when the wave has passed away, the particle is left at rest in advance of its original position at a distance equal to the total volume of the elevated water divided by the sectional area of the canal. 172. We can now examine under what circumstances the solution ex- pressed by (14) will be consistent with the assumptions made provisionally in Art. 169. The exact equation of vertical motion, viz. Dv dp gives, on integration with respect to y, fVo+n Df) p-'Po = 9p(yo + v-y)-pj -j^^y (2i) This may be replaced by the approximate equation (1), provided j3 (A + 17) be small compared with ^, where j3 denotes the maximum vertical acceleration. Now in a progressive wave, if A denote the distance between two consecutive nodes (i.e. points at which the wave-profile meets the undisturbed level), the time which the corresponding portion of the wave takes to pass a particle is A/c, and therefore the vertical velocity will be of the order lyc/A*, and the vertical acceleration of the order r)C*IX\ where r) is the maximum elevation (or depression). Hence the neglect of the vertical acceleration is justified, •provided A*/A* is a small quantity. Waves whose slope is gradual, and whose length A is large compared with the depth h of the fluid, are called 'long waves.' Again, the restriction to infinitely small motions, made in equation (3), consisted in neglecting udu/dx in comparison with du/dt. In a progressive wave we have du/dt — ± cdu/dx ; so that u must be small compared with c, and therefore, by (20), t) must be small compared with h. It is to be observed that this condition is altogether distinct from the former one, which may be legitimate in cases where the motion cannot be regarded as infinitely small. See Art. 187. * Hence, comparing with (20), we see that the ratio of the maximum vertical to the maximum horizontal velocity is of the order h/\. 171-173] Airy's Method 251 The preceding conditions will of course be satisfied in the general ciase represented by equation (14), provided they are satisfied for each of the two progressive waves into which the disturbance can be analysed. 173. There is another, although on the whole a less convenient, method of investigating the motion of Uong' waves, in which the Lagrangian plan is adopted, of making the co-ordinates refer to ttie individual particles of the fluid. For simplicity, we will consider only the case of a canal of rectangular section*. The fundamental assumption that the vertical acceleration may be neglected impUes as before that the horizontal motion of all particles in a plane perpendicidar to the length of the canal will be the same. We there- fore denote by x + f the abscissa at time t of the plane of particles whose undisturbed abscissa is x. If 77 denote the elevation of the free surface, in this plane, the equation of motion of unit breadth of a stratum whose thick- ness (in the undisturbed state) is hx will be ^*^S=-|^^ (*+'')' where the factor {dp/dx) . 8a: represents the pressure-difEerence for any two opposite particles x and x + 8a? on the two faces of the stratum, while the factor h-\- 7) represents the area of the stratum. Since we assume that the pressure about any particle depends only on its depth below the free surface we may write dx^^^dx' so that our dynamical equation is ^f--*('+'»)i <•) The equation of continuity is obtained by equating the volumes of a stratum, consisting of the same particles, in the disturbed and undisturbed conditions respectively, viz. (&x + 1^ 8x^ (h-^Tj)^ h8x, '+i-(»+i)"' <^) Between equations (1) and (2) we may eHminate either t] or (; the result in terms of f is the simpler, being or a«f dx^ a# = ^*77~5p ^^^ * Airy, Encye. Mdrop. "Tides and Waves," Art. 192 (1845); see also Stokes, "On Waves," Cawh, and Dub, Maih. Joum. t. iv. (1849) [Papers, t. ii. p. 222]. The case of a canal with sloping sides has been treated by McCowan, **0n the Theory of Long Waves. . .," PML Jliag. (5), t. xxxv. p. 250 (1892). 252 Tidal Waves [chap, vm This is the general equation of Uong' waves in a unifonn canal with vertical sides*. So far the only assumption is that the vertical acceleration of the particles may be neglected in calcidating the pressure. If we now assume, in addition, that rijh is a small quantity, the equations (2) and (3) reduce to ^--^t • (*) *^d W^^^d^ <^) The elevation tj also satisfies the equation S=^*gS (^) These are in conformity with our previous results; for the smaUness of d^/dx means that the relative displacement of any two particles is never more than a minute fraction of the distance between them, so that it is (to a first approximation) now immaterial whether the variable x be supposed to refer to a plane fixed in space, or to one moving with the fluid. 174. The potential energy of a wave, or system of waves, due to the elevation or depression of the fluid above or below the mean level is, per unit breadth, gp ilydxdy, where the integration with respect to y is to be taken between the limits and 77, and that with respect to x over the whole length of the waves. Effecting the former integration, we get \9Ph*dx (1) The kinetic energy is \ph /|*(fo (2) In a system of waves travelling in one direction only we have so that the expressions (1) and (2) are equal; or the total energy is half potential, and half kinetic. This result may be obtained in a more general manner, as foUowsf* Any progressive wave may be conceived as having been originated by the spUtting up, into two waves travelling in opposite directions, of an initial disturbance in which the particle- velocity was everywhere zero, and the energy therefore wholly potential. It appears from Art. 171 that the two derived waves are symmetrical in every respect, so that each must contain half the original store of energy. Since, however, the elevation at corresponding points is for each derived wave exactly half that of the original disturbance, the potential ♦ Airy, l.c* t Rayleigh, "On Waves," PhU, Mag. (5), t. i. p. 257 (1876) [Papers, t. L p. 251]. 173-176] Energy 253 energy of each will by (1) be one-fourth of the original store. The remaining (kinetic) part of the energy of each derived wave must therefore also be one- fourth of the original quantity. 175. If in any case of waves travelUng in one direction only, without change of form, we impress on the whole mass a velocity equal and opposite to that of propagation, the motion becomes steady ^ whilst the forces acting on any particle remain the same as before. With the help of this artifice, the laws of wave-propagation can be investigated with great ease*. Thus, in the present case we shall have, by Art. 22 (4), at the free surface, ^ = const. - 5^ (A + ^) - \q\ (1) where q is the velocity. If the slope of the wave-profile be everywhere gradual, and the depth h small compared with the length of a wave, the horizontal velocity may be taken to be uniform throughout the depth, and approximately equal to q. Hence the equation of continuity is J (A + ^) = cA, c being the velocity, in the steady motion, at places where the depth of the stream is uniform and equal to h. Substituting for q in (1), we have ? = con8t.-^A(l+|)-ic«(l + 5)-\ Hence if ry/A be small, the condition for a free surface, viz. p = const., is satisfied approximately, provided which agrees with our former result. 176. It appears from the hnearity of our equations that, in the case of sufficiently low waves, any number of independent solutions may be super- posed. For example, having given a wave of any form travelling in one direction, if we superpose its image in the plane x = 0, travelHng in the opposite direction, it is obvious that in the resulting motion the horizontal velocity will vanish at the origin, and the circumstances are therefore the same as if there were a fixed barrier at this point. We can thus understand the reflexion of a wave at a barrier; the elevations and depressions are reflected unchanged, whilst the horizontal velocity is reversed. The same results follow from the formula ^^F{ct-x)-F(ct^x)y (1) which is evidently the most general value of ^ subject to the condition that f = f or a; = 0. * Rayleigh, Z.c. I t f I 264 Tided Waves [chap, vm We can farther inyestigate without much difficulty the partial reflexion of a wave at a point where there is an abrupt change in the section of the canaL Taking the origin at the point in question, we may write, for the negative side, ,.=i^('-£)+/(*-^5). «.=J^('-S-J/('-^|) (2) and for the positive side '«=*('-6' "^^JK'-O <'> where the function F represents the original wave, and /, ^ the reflected and transmitted portions respectively. The constancy of mass requires that at the point a; =0 we should have hihitii =bj^u^, where bi, b^ are the breadths at the surface, and h^, k^ are the mean depths. We must also have at the same point 171=172, ^^ account of the continuity of pressure*. These conditions give ^ {F «) -/ («)} =^ <f> (t). F (*) +/ (0 =4, (0. We thence find that the ratios of the elevations in corresponding parts of the reflected and incident waves, and of the transmitted and incident waves, are F 6iCi+6gCj' F bjCj+bfy' ^^ respectively. The reader may easily verify that the energy contained in the reflected and transmitted waves la equal to that of the original incident wave. 177. Our investigations, so far, relate to cases oifree waves. When, in addition to gravity, small disturbing forces X, Y act on the fluid, the equation of motion is obtained as follows. We assume that within distances comparable with the depth h these forces vary only by a small fraction of their total value. On this under- standing we have, in place of Art. 169 (1), ^-^ = {9-Y){y, + r,-y) (1) and therefore i || = (<; - 7) g - (y, + , - y) g. The last term may be neglected for the reason just stated, and if we further neglect the product of the small quantities Y and drj/dx, the equation reduces to -?? = fl^ (2) pdx^dx' ^^' * It will be understood that the problem admits only of an approximate treatment, on account of the rapid change in the character of the motion near the point of discontinuity. The nature of the approximation implied in the above assumptions will become more evident if we suppose the suffixes to refer to two sections S^ and 8^, one on each side of the origin 0, at distances from which, though very small compared with the wave-length, are yet moderate multiples of the transverse dimensions of the canaL The motion of the fluid will be sensibly uniform over each of these sections, and parallel to the length. The condition in the text then expresses that there is no sensible change of level between Si and 3f, 176-178] Disturbing Fwce» 265 as before. The equation of hojrizontal motion then takes tke form ^^-'S+^. <») where X may be regarded as a function of x and t only. The equation of continuity has the same form as in Art. 169, viz. ^=-*i- ••• w Hence, on elimination of i\, w'^^w^^^ (^) • 178. The oscillations of water in a canal of uniform section, closed at both ends, may, as in the corresponding problem of Acoustics, be obtained by superposition of progressive waves travelling in opposite directions. It is more instructive, however, with a view to subsequent more difficult investi- gations, to treat the problem as an example of the general theory sketched in Art. 168. We have to determine ^so as to satisfy 8? "^S^^"^^' (^^ together with the terminal conditions that ^ = f or a; = and x = I, say. To find the free oscillations we put Z «= 0, and assume that ^ OC cos {(ft + €), where a is to be found. On substitution we obtain g+^f-» <^) whence, omitting the time-factor, f = 4 sm V B cos — . The terminal conditions give £ ^ 0, and aljc — nr, (3) where r is integral. Hence the normal mode of order r is given by * . . nrx (met , \ ... i = A^ sm -y cos [-Y- + €^j, (4) where the amplitude Af and epoch €«. are arbilarary. In the slowest oscillation (r = 1), the water sways to and fro, heaping itself up alternately at the two ends, and there is a node at the middle (x = \l). The period (2I/c) is equal to the time a progressive wave would take to traverse twice the length of the canaL 256 Tidal Waves [chap, vin The periods of the higher modes are respectively J, J, i, ... of this, but it must be remembered, in this and in other similar problems, that our theory ceases to be appUcable when the length Ijr of a semi-undulation becomes comparable with the depth h. On comparison with the general theory of Art. 168, it appears that the normal co-ordinates of the present system are quantities ji, jj, ... q^ such that when the system is displaced according to any one of them, say g^, we have > tttx f «y^sm-y-; and we infer that the most general displacement of wliich the system is capable (subject to the conditions presupposed) is given by ^ = Sy^ sin -y , (5) where ji, q%, >>. qn are arbitrary. This is in accordance with Fourier's Theorem. When expressed in terms of the normal velocities and the normal co-ordi- nates, the expressions for T and Y must reduce to sums of squares. This is easily verified, in the present case, from the formida (6). Thus if S denote the sectional area of the canal, we find 2T = pSJ i^dx = So^^2, 2V=gp^l yj^dx^ Sc^jA • • • (6) where o^ = ipSl, c^ = ir^7r^gphS/l (7) It is to be noted that, on the present reckoning, the coefficients of stability (Cr) increase with the depth. Conversely, if we assume from Fourier's Theorem that (5) is a sufficiently general expression for the value of ^ at any instant, the calculation just indicated shews that the coefficients qr are the normal co-ordinates ; and the frequencies can then be found from the general formula (10) of Art. 168; viz. we have ^r = (CrK)* = rir (5rA)*/Z (8) in agreement with (3). 179. As an example of forced waves we take the case of a imiform horizontal force X «/cos (erf -f- €) (9) This will illustrate, to a certain extent, the generation of tides in a land- locked sea of small dimensions. 178-179] Waves in a Finite Canal 257 AflnnnniTig that ^ varies as cos (<ft + e), and omitting the time-factor, the equation (1) becomes the solution of which is * / , ys . ox , « ox t=« — =4+Z)sm — h^cos — . (10) The terminal conditions give al <^hf p2» Z)sin— = (1 — COS — ) ,. (11) Hence, unless sin a\\c = 0, we have Z) =//<y* . tan (7Z/2c, so that 2/ ^ = era; . a (Z — a;) , . , . "^ sm ;c- sm — h^^ — . cos [pt + €), and 2c 2c <7 (a? - \l) . cos ((7^ + €). (12) a* cos (JaZ/c) hf oc cos ( J a(/c) u If the period of the disturbing force be large compared with that of the slowest free mode, a{/2c will be small, and the formula for the elevation becomes =.i(x^ (x — ^l) cos (at + €), (13) approximately, exactly as if the water were devoid of inertia. The horizontal displacement of the water is always in the same phase with the force, so long as the period is greater than that of the slowest free mode, or al/c < tt. If the period be diminished until it is less than the above value, the phase is reversed. When the period is exactly equal to that of a free mode of odd order (« =» 1, 3, 6, . . .), the above expressions for ^ and rj become infinite, and the solution fails. As pointed out in Art. 168, the interpretation of this is that, in the absence of dissipative forces, the ampUtude of the motion becomes so great that our fundamental approximations are no longer justified. If, on the other hand, the period coincide with that of a free mode of even order (« = 2, 4, 6, . . . ), we have sin al/c = 0, cos al/c = 1, and the terminal conditions are satisfied independently of the value of D. The forced motion may then be represented by* 2/ . ax = — -_ sin* ^- sm* ^ cos (<rf + €). (14) * In the language of the general theory, the impressed force has here no component of the particular type with which it synchronizes, so that a vibration of this tyipe is not excited at aU. In the same way a periodic pressure applied at any point of a stretched string will not excite any fundamental mode which has a node there, even though it synchronize with it. L. H. 17 258 Tid43d Waves [ohap. vra This example illustrates the fact that the effect of a disturbing force may often be conveniently calculated without resolving the force into its * normal components' (Art. 168). Another very simple case of forced oscillations, of some interest in connection with tidal theory, is that of a canal closed at one end and communicating at the other with an open sea in which a periodic oscillation 17 = a cos (cri + e) (15) is maintained. If the origin be taken at the closed end, the solution is obviously cos (axle) / . V /-./.v { denoting the length. If aljc be small the tide has sensibly the same amplitude at all points of the canal. For particular values of I (determined by cos aljc = 0) the solution fails through the ampUtude becoming infinite. Canal Theory of the Tides. 180. The theory of forced osciUations in canals, or on open sheets of water, owes most of its interest to its bearing on the phenomena of the tides. The * canal theory,' in particular, has been treated very fully by Airy*. We will consider one or two of the more interesting problems. The calculation of the disturbing effect of a distant body on the waters of the ocean is placed for convenience in an Appendix at the end of this Chapter. It appears that the disturbing effect of the moon, for example, at a point P of the earth's surface, may be represented by. a potential €l whose approximate value is « = f^(i-C08«^). (1) where M denotes the mass of the moon, D its distance from the earth's centre, a the earth's radius, y the * constant of gravitation,' and ^ the moon's zenith distance at the place P. This gives a horizontal acceleration dn/ad^, or . /sin2a, .(2) towards the point of the earth's surface which is vertically beneath the moon, where D^ (3) * EjicgcL Mdrop, *' Tides and Waves," Section vi. (1845). Several of the leading features of the theory had been made out, by very simple methods, by Young, in 1813 and 1823 [Works, t. ii. pp. 262. 291]. 179-181] Canal Theory of the Tides 259 If E be the earth's mass, we may write g » yE/a^, whence / 3 M /a\8 g^2' E '[dJ ' Putting M/E = ^, a/Z) = ^, this gives f/g = 8-57 x lO-®. When the sun is the disturbing body, the corresponding ratio iaf/g = 3*78 x 10~®. It is convenient, for some purposes, to introduce a Unear magnitude £r, defined by H'-afig (4) If we put a == 21 X 10* feet, this gives, for the lunar tide, H = 1*80 ft., and for the solar tide H = '79 ft. It is shewn in the Appendix that H measures the maximum range of the tide, from high water to low water, on the * equi- librium theory.' 181. Take now the case of a imiform canal coincident with the earth's equator, and let us suppose for simplicity that the moon describes a circular orbit in the same plane. Let ^ be the displacement, relative to the earth's surface, of a particle of water whose mean position is in longitude </}, measured eastwards from some fixed meridian. If co be the angular velocity of the earth's rotation, the actual displacement of the particle at time t will be f + «a>f, so that the tangential acceleration will be d^^/dfi. If we suppose the 'centrifugal force' to be as usual allowed for in the value of g, the processes of Arts. 169, 177 will apply without further alteration. If n denote the angular velocity of the moon westward, relative to the fixed meridian*, we may write in Art. 180 (2) a^ = w« + ^ + €, so that the equation of motion is P = ''*a-|f«--^«^2(««+^ + e) (1) The free oscillations are determined by the consideration that ^ is necessarily a periodic function of ^, its value recurring whenever <f> increases by 2n. It may therefore be expressed, by Fourier's Theorem, in the form I = S (P^cosr^ + g^sinr<^) (2) Substituting in (1), with the last term omitted, it i^' found that P^ and Q^ must satisfy the equation dip «.2^2 The motion, in any normal mode, is therefore simple-harmonic, of period 27ralrc. * That is, nsw-Tii^ifn^be the angular velocity of the moon in her orbit. 17—2 260 Tidal Waves [ohap. vin For the forced waves, or tides, we find ^--i^^^2^^^(^ + <f>-^^)^ W whence iy = ^ c« - n^a^ cos 2 (nt + ^ + c) (5) The tide is therefore semi-diurnal (the lunar day being of course understood), and is 'direct' or 'inverted,' i.e. there is high or low water beneath the moon, according as c ^ na, in other words according as the velocity, relative to the earth's surface, of a point which moves so as to be always vertically beneath the moon, is less or greater than that of a free wave. In the actual case of the earth we have n^a^ n^a a a so that unless the depth of the canal were to greatly exceed such depths as actually occur in the ocean, the tides would be inverted. This residt, which is sometimes felt as a paradox, comes under a general principle referred to in Art. 168. It is a consequence of the comparative slowness of the free (i^illations in an equatorial canal of moderate depth. It appears from the rough numerical table on p. 249 that with a depth of 11250 feet a free wave would take about 30 hours to describe the earth's semi-circumference, whereas the period of the tidal disturbing force is only a little over 12 hours. The formida (5) is, in fact, a particular case of Art. 168 (14), for it may be written where rj is the elevation given by the 'equilibrium theory,' viz. ^ = Jff cos 2 (n^ -f ^ -f- €), (7) and a =» 2n, Gq = 2c/a. For such moderate depths as 10000 feet and under, n^a^ is large com- pared with gh ; the amphtude of the horizontal motion, as given by (4), is then //4n2, or gj^rt^a . £f, nearly, being approximately independent of the depth. In the case of the limar tide this amplitude is about 140 feet. The maximum elevation is obtained by multiplying by 2%/a; this gives, for a depth of 10000 feet, a height of only 133 of a foot. For greater depths the tides would be higher, but still inverted, until we reach the critical depth n^a^/g, which is about 13 miles. For depths beyond this limit, the tides become direct, and approximate more and more to the value given by the equilibrium theory*. ♦ Cf. Young, Lc. ante p. 2C8. 181-183] Tide in Equatorial Canal 261 182. The case of a circular canal parallel to the equator can be worked out in a similar manner. If the moon's orbit be still supposed to he in the plane of the equator, we find by spherical trigonometry cos ^ = sin cos (n^ + <^ + €), (1) where is the co-latitude, and <f> the longitude. The disturbing force in longitude is therefore ^-^^ = -/sin flsin 2{nt + d>-{-€) (2) Thisleadsto ^ = J ^^-^^^1^^^ (3) Hence if wa > c the tide will be direct or inverted according as 5 $ sin"* c\na. If the depth be so great that ona^ the tides will be direct for all values of 0. If the moon be not in the plane of the equator, but have a co-declination A, the formula (1) is replaced by cos & = cos ^ cos A + sin sin A cos a, (4) where a is the hour-angle of the moon from the meridian of P. For simplicity, we will neglect the moon's motion in declination in comparison with the earth's angular velocity of rotation ; thus we put a » «i + ^ + €, and treat A as constant. The resulting expression for the disturbing force along the parallel is found to be oiii = —/cos 6 sin 2A sin (n* + ^ + €) -/sin Q sin« A sin 2 (n< + ^ + «) (5) We thence obtain 17 = i -^ ^ , . o -qSU^ 20 sin 2A cos (n^ + <A + c) + \ 1 , o . .^ sin* e sin« A cos 2 (n^ + <^ + €) (6) The first term gives a 'diurnal' tide of period %T\n\ this vanishes and changes sign when the moon crosses the equator, i,e, twice a month. The second term represents a semi-diurnal tide of period 7r/n, whose ampUtude is now less than before in the ratio of sin* A to 1. 183. In the case of a canal coincident with a meridian we should have to take account of the fact that the undisturbed figure of the free surface is one of relative equilibrium under gravity and centrifugal force, and is therefore not exactly circular. We shall have occasion later on to treat the question of displacements relative to a rotating globe somewhat carefully; for the present we will assume by anticipation that in a narrow canal the 262 Tidal Waves [chap, vm disturbances are sensibly the same as if the earth were at rest, and the disturbing body were to revolve round it with the proper relative motion. If the moon be supposed to move in the plane of the equator, the hour- angle from the meridian of the canal may be denoted by w^ + €, and if d be the co-latitude of any point P on the canal, we find cos Sr = sin . cos (n^ + c) (1) The equation of motion is therefore Solving, we find T? = - iff cos 2^ - i -^ 2—2 ^^8 25 • ^^ 2 (n« + €) (3) The first term represents a permanent change of mean level to the extent 7^ = - Jffcos2e (4) The fluctuations above and below the disturbed mean level are given by the second term in (3). This represents a semi-diurnal tide ; and we notice that if, as in the actual case of the earth, c be less than na, there will be high water in latitudes above 45°, and low water in latitudes below 45°, when the moon is in the meridian of the canal, and vice versa when the moon is 90° from that meridian. These circumstances would be all reversed if c were greater than na. When the moon is not on the equator, but has a given declination, the mean level, as indicated by the term corresponding to (4), has a coefficient depending on the declination, and the consequent variations in it indicate a fortnightly (or, in the case of the sun, a semi-annual) tide. There is also introduced a diurnal tide whose sign depends on the declination. The reader will have no difficulty in examining these points, by means of the gene];al value of ii given in the Appendix. • 184. In the case of a uniform canal encircling the globe (Arts. 181, 182) there is necessarily everywhere exact agreement (or exact opposition) of phase between the tidal elevation and the forces which generate it. This no longer holds, however, in the case of a canal or ocean of limited extent. Let us take for instance the case of an equatorial canal of finite length. If the origin of time be suitably chosen we have 3=''*^*--^«^'^2(«^+^). (1) with the condition that £ = at the ends, where ^ = ± a, say. 183-184] Tid^ in Finite Canal 263 If we neglect the inertia of the water the term d^i/dfi is to be omitted, and we find f = J'^-]8in2w^cos2a + -cos2w^sin2a — sin 2 (n< + ^)[. ..(2) Hence 17 = - *|| = Ji? |cos2 (n^ + <^) - ?^cos2n«l, (3) where H ^^fa/g, as in Art. 180. This is the elevation on the (corrected) 'equiUbrium' theory referred to in the Appendix to this Chapter. At the centre {<f> = 0) of the canal we have ,, = Jffcofl2««(l-^) (4) If a be small the range is here very small, but there is not a node in the absolute sense of the term. The times of high water coincide with the transits of moon and * anti-moon.'* At the ends ^ = ± a we have 71 = iff |(l -^-^^) cos2 (n<± a)T ^'^^^ Bm2{nt±a) = iHRo cos 2 (n« ± a =F cq), (5) sin 4a » • « 1 — cos 4a 4^' /?o8m2eo= ^ if Rq cos 2€o = 1 z — , Rq sm 2^0 = r (^) Here €0 denotes the hour-angle of the moon W. of the meridian when there is high water at the eastern end of the canal, or E. of the meridian when there is Idgh water at the western end. When a is small we have Bo = 2a, €o=-iir + |a, (7) approximately. When the inertia of the water is taken into account we have f = L o IV a I sin 2 {nt + ^) - -=-4 — {sin 2(fU + a) sin 2m {<t> + a) — sin 2 (w< — a) sin 2m (^ — a)} , (8) where m = na/c. Hence f ■n = — 1 -X r COS 2int-\-6) r—T- {sin 2 (n^ + a) cos 2m (tk + a) ' ^ m* — 1 L sm 4ma ^^ . ' — sin 2 (w< — a) cos 2m (^ — a)} (9) If we imagine w to tend to the limit we obtain the formula (3) of the equilibrium theory. It may be noticed that the expressions do not become * This term is explained in the Appendix to this Chapter. t Of. Airy, "Tides and Waves," Art. 301. The discussion in the text is from a paper in the Phil Mag, (6), t. xxix. p. 737 (1915). 264 Tidal Waves [chap, vm infinite f or m -^ 1 as they do in the case of an endless canal. In all cases which are at all comparable with oceanic conditions m is, however, considerably greater than unity. At the centre of the canal we have H o ^ /n m sin 2a\ cos 2nt 1 1 ; — s — \ sin2ma/ ,=-j (10) m* — 1 \ sm As in the eqniUbrium theory, the range is very smaU if a be small, but there is not a true node. At the ends we find H "^^^W^^ m sin 4a \ sin 4ma — ljcos2 (vJt± a) m (cos ima — cos 4a) . « , ^ . x sm 4ma if Bi cos 2€i = (11) = \HR^ cos 2 (n^ ± a qpci), m sin 4a — sin 4ma ^ . ^ m (cos 4ma — cos 4a) .-^v , Ri sm 2€i = -7—= — -. . . . . .(12) (m* — 1) sm 4ma (13) (m* — 1) sin 4ma When a \a small we have Ri = 2a, €1 = — Jtt + fa, approximately, as in the case of the equilibrium theory. The value of Ri becomes infinite when sin 4ma » 0. This determines the critical lengths of the canal for which there is a free period equal to 7r/n, or half a lunar day. The limiting value of c^ in such a case is given by tan 2€i = — cot 2a, or = tan 2a, according as 4ma \s an odd or even multiple of tt. • Corrected Equilibrium Theory Dynamical Theory 2a 2aa Range at Range at fo Range at Range at 1 €1 (degrees) (miles) centre ends (degrees) centre ends (degrees) -45 -45 9 540 •004 •167 -42 •004 •166 -41-9 18 1080 •016 •311 -39 •018 •396 -38-5 27 1620 •037 •460 -36 •044 •941 -33-9 31-5 1890 •050 •531 -34-5 •063 1946 -30-9 36 2160 •065 •601 -33 •089 00 (-27 \+63 + 68-2 40-5 2430 •081 668 -31-6 •125 1-956 45 2700 •100 •733 -301 •174 •987 + 75-7 54 3240 •142 •853 -27-2 •354 •660 -83-5 63 3780 •190 •969 -24 4 •918 1141 -65-1 72 4320 •243 1051 -216 00 00 j -54 \+36 + 44 5 81 4860 •301 M27 -18-9 h459 1112 90 6400 •363 M85 -162 •864 •613 + 55-9 184-rl85] Canal of Varying Section 265 The table illustratee the ease of m =2*5. If irjn — 12 lunar hours this implies a depth of 10820 ft., which is of the same order of magnitude as the mean depth of the ocean. The corresponding wave-velooity is about 360 sea-miles per hour. The first critical length is 2160 miles (a =t\r^)- ^^® ^^^ ^ terms of which the range is expressed is the quantity H, whose value for the lunar tide is about 1*80 ft. The hour-angles cq and f^ are adjusted so as to lie always between ±: 90°, and the positive sign indicates position W. of the meridian in the case of the eastern end of the canal, and E. of the meridian for the western end. Wave-Motion in a Canal of Variable Section, 185. When the section (S, say) of the canal is not unifonn, but varies gradually from point to point, the equation of continuity is, by Art. 169 (11), where b denotes the breadth at the surface. If h denote the mean depth over the width .6, we have S = bh, and therefore ^--lrx<'^^^' (2) where h, b are now functions of z. The dynamical equation has the same form as before, viz. w=-nx (^) Between (2) and (3) we may eliminate either ry or f ; the equation in 17 is ^=\U«'l) <*) The laws of propagation of waves in a canal of gradually varying rect- angular section were investigated by Green*. His results, freed from the restriction to the special form of section, may be obtained as follows. If we introduce a variable r defined by Tr-(9^^ W in place of x, the equation (4) transforms into where the accents denote differentiations with respect to t. If 6 and h were constcuits, the equation would be satisfied hy rj = F {t - 1), aa ia. Art. 170 ; in the present case we assume, for trial, Ti=e.F(T-t), (7) where 6 is a function of r only. Substituting in (6), we find ^e' F' e" fh' \hr\(F' e'\ ^ ,«. • "On the Motion of Waves in a Variable Canal of small depth and width," Camb. Trans. t. vi. (1837) [Papers, p. 225]; see also Airy, "Tides and Waves," Art. 2«0. 266 Tidai Waves [chap, vin The terms of this which involve F will cancel provided or e = 06 " i A ~ i ( 9 ) C being a constant. Hence, provided the remaining terms in (8) may be neglected, the equation (4) will be satisfied. The above approximation is justified, provided we can neglect 0"lB' and 676 in com- parison with F'/F. As regards 676, it appears from (9) and (7) that this is equivalent to neglecting b~^ . db/dx and h~^ . c^fdx in comparison with rj"^ . drj/dz. If, now, X denote a wave-length, in the general sense of Art. 172, dri/dx is of the order 17/X, so that the assump- tion in question is that Tidb/dx and ^dh/dx are small compared with b and h, respectively. In other words, it is assumed that the transverse dimensions of the canal vary only by small fractions of themselves within the limits of a wave-length. It is easily seen, in like manner, that the neglect of 6^7^' ^ comparison with F'/F implies a similar limitation to the rates of change of db/dx and dh/dx. Since the equation (4) is unaltered when we reverse the sign of t, the complete solution, subject to the above restrictions, is ,; =6 -4 A-i {^ (r - +/(r +t)} (10) where F and / are arbitrary functions. The first term in this represents a wave travelling in the direction of x-positive ; the velocity of propagation past any point is determined by the consideration that any particular phase is recovered when br and lit have equal values, and is therefore equal to *J{gk), by (5), as we should expect from the case of a uniform section. In like manner the second term in (10) represents a wave travelling in the direction of ar-negative. In each case the elevation of any particular part of the wave alters, as it proceeds, according to the law b-h-i. The reflection of a progressive wave at a point where the section of a canal suddenly changes has been considered in Art. 176. The formulae there given shew, as we should expect, that the smaller the change in the dimensions of the section, the smaller will be the amplitude of the reflected wave. The case where the change from one section to the other is continuous, instead of abrupt, has been investigated by Kayleigh for a special law of transition*. It appears that if the space within which the transition is completed be a moderate multiple of a wave-length there is practically no reflection; whilst in the opposite extreme the results agree with those of Art. 176. If we assume, on the ba.sis of these results, that when the change of section within a wave-length may be neglected a progressive wave suffers no appreciable disintegration by reflection, the law of amplitude easily follows from the principle of energy f. It appears from Art. 174 that the energy of * "On Reflection of Vibrations at the Confines of two Media between which the Transition is gradual," Proc, Lond, Math, Soc. t. zi. p. 51 (1880) [Papers, t. i. p. 460]; Theory of Sound, 2nd^., London, 1894, Art 1486. t Rayleigh, Ic. anU p. 262. 186-186] Canal of Varying Section 267 the wave varies as the length, the breadth, and the square of the height, and it is easily seen that the length of the wave, in different parts of the canal, varies as the corresponding velocity of propagation, and therefore as the square root of the mean depth. Hence, in the above notation, ifbhr is constant, or which is Green's law above found. 186. In the case of simple harmonic motion, where t] oc cos (at + e). the equation (4) of the preceding Art. becomes f5(»»l)+-'-« m Some particular cases of considerable interest can be solved with ease. 1^. For example, let us take the case of a canal whose breadth varies as the distance from the end a; = 0, the depth being uniform ; and let us suppose that at its mouth (x = a) the canal communicates with an open sea in which a tidal oscillation 7^ = C cos ((7« + €) (2) is maintained. Putting h = const., 6 oc x, in (1), we find D+il+'''-» '" provided A;* = o^jgh (4) Hence -n = C / L , cos (a< + c) (5) Jo(*a) The curve y ^^ Jq (x) is figured on p. 278 ; it indicates how the amplitude of the forced oscillation increases, whilst the wave-length is practically constant, a.s we proceed up the canal from the mouth. 2^. Let us suppose that the variation is in the depth only, and that this increases uniformly from the end x = of the canal to the mouth, the remain- ing circumstances being as before. If, in (1), we put h = hQx/a, k = ci^ajgh^y we obtain i('i)+''-» <" The solution of this which is finite for x = is . /- KX K^X^ \ .„. r) = All — Y^ -{- ^2 22 "~ • • • ] » I ' / or 7] = AJo (2/c*x*), (8) whence finally, restoring the time-factor and determining the constant, = 0«M?!L?) cos (<H + €) (9) 268 Tidal Waves [chap, vin The annexed diagram of the curve y = Jq (V^), where, for clearness, the scale adopted for y is 200 times that of x, shews how the amplitude continually increases, and the wave-length diminishes, as we travel up the canal. These examples may serve to illustrate the exaggeration of oceanic tides which takes place in shallow seas and in estuaries. We add one or two simple problems of free oscillations. 3^. Let us take the case of a canal of uniform breadth, of length 2a, whose bed slopes uniformly from either end to the middle. If we take the origin at one end, the motion in the first half of the canal will be determined, as above, by 7? = 4 Jo (2#cM), (10) where k ^ o^djgh^^ h^ denoting the depth at the middle. It is evident that the normal modes will fall into two classes. In the first of these 77 will have opposite values at corresponding points of the two halves of the canal, and will therefore vanish at the centre (x = a). The values of a are then determined by Jo(2A*) = 0, (11) viz. #c being any root of this, we have „^i9j^,^^a)i (12) In the second class, the value of t) is symmetrical with respect to the centre, so that dri/dx = at the middle. This gives Jo'(2A*) = (13) It appears that the slowest oscillation is of the asymmetrical class, and corresponds to the smallest root of (11), which is 2ic*a* = -TeSSw, whence ^ . 1-306 X -*^. 186-187] Canal of Varying Section 269 4^. Again, let us suppose that the depth of the canal varies according to the law A = Ao(l-^), (14) where x now denotes the distance from the middle. Substituting in (1), with 6 = const., we find 4{('-s:)l}-£'-» <>») If we put ff» = n (n + 1)^, (16) this is of the same form a.s the general equation of zonal harmonics, Art. 84 (1). In the present problem n is determined by the condition that 77 must be finite for xja = ± 1. This requires (Art. 85) that n should be integral; the normal modes are therefore of the type = CP, (?) . cos M -h €), (17) where P„ is a zonal harmonic, the value of a being determined by (16). In the slowest oscillation (n = 1), the profile of the free surface is a straight line. For a canal of uniform depth h^, and of the same length (2a), the corresponding value of a would be 7rc/2a, where c = {gh^ , Hence in the present case the frequency is less, in the ratio 2\/27r, or -9003*. The forced oscillations due to a uniform disturbing force X =/cos ((7« + e) (18) can be obtained by the rule of Art. 168 (14). The equilibrium form of the free surface is evidently ^ =-^ X cos (a« -h €), (19) and, since the given force is of the normal type w = 1, we have ^ = (y (1 -Vo') '^ '^"^ <°* + '^' (20) where ag* = ^gh^ja!^. Waves of Finite Amplitude. 187. When the elevation i) is not small compared with the mean depth A, waves, even in an uniform canal of rectangular section, are no longer propagated without change of type. The question was first investigated by * For extenBions, and applications to the theory of * seiches* in lochs, see Chrystal, "Some Results in the Mathematical Theory of Seiches," Proc. R. S. Edin. t. xxv. p. 328 (1904), and Tran9, R, 8, Edin, t. xli. p. 699 (1906). I I 270 Tidal Waves [ohap. vra Airy*, by methods of successive approximation. He found that in a pro* gressive wave different parts will travel with different velocities, the wave- velocity corresponding to an elevation 77 being given approximately by »(1H-Ii), where c is the velocity corresponding to infinitely small amplitude. A more complete view of the matter can be obtained by the method employed by Biemann in treating the analogous problem in Acoustics, to which reference will be made in Chapter x. The sole assumption on which we are now proceeding is that the vertical acceleration may be neglected. It follows, as explained in Art. 168, that the horizontal velocity may be taken to be uniform over any section of the canal. The djmamical equation is du , du dri ¥ + «^=-^ai' (1) as before, and the equation of continuity, in the case of a rectangular section, is easily seen to be a^P + '?)«}=-!. ; (2) where h is the depth. This may be written | + «^--(» + ,)g (3) Let us now write P =/(!,) + M, Q =/(,,) -u, (4) where the function / (rj) is as yet at our disposal. If we multiply (3) by /' (tj), and add to (1), we get If we now determine/ (tj) so that {h + v){f'm* = 9 (5) this may be written ¥ + «I=-(* + ''>/'('')S («) In the same way we find f + .g- (»+,)/'wg m The condition (5) is satisfied by /(,) = 2c|(l + |)*-lj. (8) ♦ "Tides and Waves," Art. 198. 187] Waves of Finite Amplitude 271 where c^ {g^) - Tbe arbitrary constant has been chosen so as to make P and Q vanish in the parts of the canal which are free from disturbance, but this is not essential. Substituting in (6) and (7) we find tf-[&-{.(.+i/+„}*]g,^ V . ^«^^ *.[&+jc(l+|)»-„}*]|. It appears, therefore, that dP = 0, i,e, P is constant, for a geometrical point moving with the velocity S-('^l)*+». w whilst Q is constant for a point moving with the velocity i--«('-^i)'^« <"' Hence any given value of P travels forwards, and any given value of Q travels backwards, with the velocities given by (10) and (11) respectively. The values of P and Q are determined by those of i] and w, and conversely. As an example, let us suppose that the initial disturbance is confined to the space for which a<x<h, so that P and Q are initially zero for X <a and x>h. The region within which P differs from zero therefore advances, whilst that within which Q differs from zero recedes, so that after a time these regions separate, and leave between them a space within which P = 0, = 0, and the fluid is therefore at rest. The original disturbance has now been resolved into two progressive waves travelling in opposite directions. In the advancing wave we have = 0, iP = i* = 2c{(l+|)*-lj,.... (12) so that the elevation and the particle- velocity are connected by a definite relation (cf. Art. 171). The wave-velocity is given by (10) and (12), viz. it is» c H'+D'-^l <") To the first order of t^/A, this is in agreement with Airy's result. Similar conclusions can be drawn in regard to the receding wave*. Since the wave- velocity increases with the elevation, it appears that in a progressive wave-system the slopes will become continually steeper in front, and more gradual behind, until at length a state of things is reached in * The above reeulta can also be deduced from the equation (3) of Art. 173, to which RiemaDD*8 method can readily be adapted. 272 Tidal Waves [chap, vni which we are no longer justified in neglecting the vertical acceleration. As to what happens after this point we have at present no guide from theory; observation shews, however, that the crests tend ultimately to curl over and break. The case of a *bore,' where there is a transition from one uniform level to another, may be investigated by the artifice of steady motion (Art. 175). If Q denote the volume per unit breadth which crosses each section in unit time we have Uyh^ = u^h^ = e, (14) where the sufiGlxes refer to the two uniform states, h^ and A^ denoting the depths. Considering the mass of fluid which is at a given instant contained between two cross-sections, one on each side of the transition, we see that in unit time it gains momentum to the amount pQ (u^ — i^), the second section being supposed to lie to the right of the first. Since the mean pressures over the sections are ^gphi and ^gph^, we have Q {«. - «i) = isr (^ - A,) (15) Hence, and from (14), Q» = yh^ht ih + ht) (16) If we impress on everything a velocity — Mj we get the case of a wave invading still water with a velocity of propagation «! = V^f 2., J (") in the negative direction. The particle-velocity in the advancing wave is Wj — Mj in the direction of propagation. This is positive or negative according as ^2 < ^1) ^•^* according as the wave is one of elevation or depression. The equation of energy is however violated, imless the difference of level be regarded as infinitesimal. If, in the steady motion, we consider a particle moving along the surface stream-line, its loss of energy in passing the place of transition is ip W - V) -\-9Pih-K) (18) per unit volume. In virtue of (14) and (16) this takes the form gp (h^ - Ai)« 4A1A3 ^^^^ Hence, so far as this investigation goes, a bore of elevation {h^ > hi) can be propagated unchanged on the assumption that dissipation of energy takes place to a suitable extent at the transition. If however ^2 < ^1 > ^^^ expression. (19) is negative, and a supply of energy would be necessary. It follows that a negative bore of finite height cannot in any case travel unchanged*. ♦ Rayleigh, " On the Theory of Long Waves and Bores," Proc, Roy, Soc. A, t. xo. p. 324 (1914). 187-188] Tides of Second Order 273 188. In the detailed application of the equations (1) and (3) to tidal phenomena, it is usual to follow the method of successive approximation. As an example, we will take the case of a canal communicating at one end {x = 0) with an open sea, where the elevation is given by 7^ = a cos at (20) For a first approximation we have ¥-"^S' ¥-"^^ ^^^' the solution of whioh, consistent with (20), is rf=a COR alt J, u=: — 0OB<r(t ) (22) For a second approximation we substitute these values of rj and i^ in (1) and (3), and obtain ¥ = -^S-V'^2<r^t--j. ^ = _A^-^Bm2<r(«--j. ...(23) Integrating these by the usual methods, we find, as the solution consistent with (20), y • • • • I get U = ^— cos cr c (..?)-i^%os2.(.-f)-i?!^%sin2.(.^?).J (24) The annexed figure shews, with, of course, exaggerated amplitude, the profile of the waves in a particular case, as determined by the first of these equations. It is to be noted that if we fix our attention on a particular point of the canal, the rise and fall of the water do not take place symmetrically, the fall occupying a longer time than the rise. The occurrence of the factor x outside trigonometrical terms in (24) shews that there is a Hmit beyond which the approximation breaks down. The condition for the success of the approximation is evidently that gtrax/c^ should be small. Putting c*=gh, X=29rc/o', this fraction becomes equal to 2ir (a/h) . (x/X). Hence however small the ratio of the original elevation (a) to the depth, the fraction ceases to be small when x is a sufficient multiple of the wave-length (X). It is to be noticed that the Hmit here indicated is already being overstepped in the right-hand portions of the figure; and that the peculiar features which are beginning to shew themselves on the rear slope are an indication rather of the imperfections of the analysis than of any actual property of the waves. If we were to trace the curve further, we should find a secondary maximum and minimum of elevation developing themselves on the rear slope. In this way Airy attempted to explain the phenomenon of a double high-water which is observed in some rivers; but, for the reason given, the argument cannot be sustained *. The same difficulty does not necessarily present itself in the case of a canal closed by a fixed barrier at a distance from the mouth, or, again, in the case of the forced waves due to * MoCowan, Ix. ante p. 251. L. H. .18 274 Tided Waves [chap, vra a periodio horizontal force in a canal dosed at both ends (Art. 179). Enough has, however, been giv^ to shew the general character of the results to be expected in such cases. For further details we must refer to Airy*s treatise*. When analysed, as jn (24), into a series of simple-harmonic functions of the time, the expression for the elevation of the water at any particular place (x) consists of two terms, of which the second represents an 'over- tide,' or 'tide of the second order,' being propoiv tional to a'; its frequency is double that of the primary disturbance (20). If we were to continue the approximation we should obtain tides of higher orders, whose frequencies are 3, 4, . . . times that of the primary. If, in place of (20), the disturbance at the mouth of the canal were given by f =a cos ai'\-a' cos (o-'t + c), it is easily seen that in the second approximation we should in like manner obtain tides of periods 2w/(o- + a) and 27r/(o' — cr') ; these are called 'compound tides.' They are analogous to the 'combination- tones' in Acoustics which were first investigated by Helmholtzf. Propagation in Two Dimensions. 189. Let us suppose, in the first instance, that we have a plane sheet of water of uniform depth h. If the vertical acceleration be neglected, the horizontal motion will as before be the same for all particles in the same vertical line. The axes of x, y being horizontal, let w, v be the component horizontal velocities at the point (x, y), and let ^ be the corresponding elevation of the free surface above the undisturbed level. The equation of continuity may be obtained by calculating the flip: of matter into the columnar space which stands on the elementary rectangle hxhy ; viz. we have, neglecting terms of the second order, ^ (uhhy) 8x + U^hhx) 8y = - I {(J + h) 8x8y}, ^^«°^« i = -Hl+|) (1) The dynamical equations are, in the absence of disturbing forces, du _^ dp 3v _ dp where we may write if Zo denote the ordinate of the free surface in the undisturbed state. We thus obtain du _ 9 J 3v _ 3J ¥"*"**ai' a?"""^a^ ^^^ ♦ "Tides and Waves," Arts. 198, ... and 308. See also G. H. Darwin, "Tides," Encyc. Britann, (9th ed.) t. xziii. pp. 362, 363 (1888). t "Ueber Combinationstone," Berl Monatsber, May 22, 1866 [Wiss. Ahh. t. i. p. 266]; and "Theorie der Luftsohwingungen in Rohren mit offenen Enden," CreUe, t. Ivii. p. 14 (1869) [Wiss. Abh. t. i. p. 318]. 188-189] Waves on an Open Sheet of Water 276 If we eUminate u and v, we find where c^ = gh ba before. In the application to simple-hannonic motion, the equations are shortened if we assume a complex time-factor «* <'*+•', and reject, in the end, the imaginary parts of our expressions. This is legitimate so long as we have to deal solely with linear equations. We have then, from (2), « = ^P. « = *^|-^ (4) a ox' a dy ^ ' whilst (3) becomes 3^ + 3^ + ^^"^' ^^) where h^ = a^/c^ (6) The condition to be satisfied at a vertical bounding wall is obtained at once from (4), viz. it is i-». <') if Sn denote an element of the normal to the boundary. When the fluid is subject to small disturbing forces whose variation within the limits of the depth may be neglected, the equations (2) are replaced by at"" ^dx dx' dt~ ^dy dy' ^^ where CI is the potential of these forces. If we put f = - Q/^, (9) 80 that ^ denotes the equilibrium-elevation corresponding to the potential Q, these may be written s-4«-a s-4«-fi "») In the case of simple-harmonic motion, these take the forms "-!s«-o. "-H«-" <"» whence, substituting in the equation of continuity (1), we obtain ( V + *«) C = V,af, (12) ^ ^^*=^ + aT- (1') 18—2 276 Tidal Waves [ohap. vm and 1^ = a^lgh, ba before. The condition to be satisfied at a vertical boundary is now a|(^-?) = o (1*) 190. The equation (3) of Art. 189 is identical in form with that which presents itself in the theory of the transverse vibrations of a uniformly stretched membrane. A still closer analogy, when regard -is had to the boundary-conditions, is furnished by the theory of cylindrical waves of sound*. Indeed many of the results obtained in this latter theory can be at once transferred to our present subject. Thus, to find the free oscillations of a sheet of water bounded by vertical walls, we require a solution of (V,« + A»)S = (1) subject to the boundary-condition g-» p) Just as in Art. 178 it will be found that such a solution is possible only for certain values of k, which accordingly determine the periods (^n/kc) of the various normal modes. Thus, in the case of a rectangular boundary, if we take the origin at one corner, and the axes of x, y along two of the sides, the boundary-conditions are that dt^jdx = for x = and x = a, and dl,jdy = f or y = and y = b, where a, 6 are the lengths of the edges parallel to x, y respectively. The general value of J subject to these conditions is given by the double Fourier's series i = SSil„,„ COS -^ cos -^, (3) where the summations include all integral values of m, n from to oo » Substituting in (1) we find If a > 6, the component oscillation of longest period is got by making w = 1, n = 0, whence ka = n. The motion is then everywhere parallel to the longer side of the rectangle. Cf. Art. 178. 191. In the case of a circular sheet of water, it is convenient to take the origin at the centre, and to transform to polar co-ordinates, writing x = r cos 0y y = rsiad. ♦ Rayleigh, Theory of Sound, Art. 339. 189-191] Circular Basin 277 The equation (1) of the preceding Art. becomes dr^^rd^^V^W'^''^ ^ (^) This might of course have been established independently. As regards dependence on 0, the value of J may, by Fourier's Theorem, be supposed expanded in a series of cosines and sines of multiples of ; we thus obtain a series of terms of the form /(^)IV <') It is found on substitution in (1) that each of these terms must satisfy the equation independently, and that rir) + lf'ir) + (h*-^f{r)^0 (3) This is of the same form as Art. 101 (14). Since f must be finite for r = 0, the various normal modes are given by cos"" J = A,J, (hr) . y8d,coB{at + €), (4) where 8 may have any of the values 0, 1, 2, 3, ... , and A, is an arbitrary constant. The admissible values of k are determined by the condition that d^/dr = at the boundary r = a, say, or J/{ka)=^0 (5) The corresponding 'speeds' (a) of the oscillations are then given by In the ca.se 8 — 0, the motion is symmetrical about the origin, so that the waves have annular ridges and furrows. The lowest roots of Jo' (*a) = 0, or Ji (*a) = 0, (6) are given by Aw/tt = 1-2197, 2-2330, 3-2383, . . . , (7) these values tending ultimately to the form ka/'jr = m-\- 1, where m is integral*. In the mth mode of the symmetrical class there are m nodal circles whose radii are given by J = or Jo(*^) = (8) The roots of thisf are jfcr/TT = -7656, 1-7571, 2-7546, (9) * Stokes, **0n the Numerioal Galoulation of a class of Definite Integrals and Infinite Series," Comb. Trans, t. ix. (1850) [Papere, t. ii. p. 355]. It is to be noticed that kajv is equal to tJt, where r is the actual period, and Tq is the time a progressive wave would take to travel with the velocity *J{gh) over a space equal to the diameter 2a. t Stokes, ^e. 278 Tidal Waves [chap, vra I o Ik S JS 191] Circular Basin 279 For example, in the first symmetrical mode there is one nodal circle r = •628a. The form of the section of the free surface by a plane through the axis of 2, in any of these modes, will be understood from the drawing of the curve y = •'^0 (^)> which is given on the preceding page. When 8>Q there are 8 equidistant nodal diameters, in addition to the nodal circles J.(ifcr) = (10) It is to be noticed that, owing to the equality of the frequencies of the two modes represented by (4), the normal modes are now to a certain extent indeterminate; viz. in place of cos«0 or Ansd we might substitute cos 8(0 — a,), where a, is arbitrary. The nodal diameters are then given by a 2m+l . . ^ - a, = — 27~ ^' *• (^^) where m = 0, 1, 2, . . . , 5 — 1. The indeterminateness disappears, and the frequencies become unequal, if the boundary deviate, however slightly, from the circular form. In the case of the circular boundary, we obtain by superposition of two fundamental modes of the same period, in different phases, a solution S = C,J, (Jfcr) . cos (at T «fl + 6) (12) This represents a system of waves travelling unchanged round the origin with an angular velocity a/« in the positive or negative direction of 0. The motion of the individual particles is easily seen from Art. 189 (4) to be elliptic-harmonic, one principal axis of each elliptic orbit being along the radius vector. All this is in accordance with the general theory referred to in Art. 168. The most interesting modes of the unsymmetrical class are those corre- sponding to « = 1, e,g, t, = AJ-^ (hr) cos . cos (ai -h e), (13) where k is determined by J/(Jfca) = (14) The roots of this are* Jfca/7r='586, 1-697, 2-717*, (15) We have now one nodal diameter (6 = Jtt), whose position is, however, indeterminate, since the origin of is arbitrary. In the corresponding modes for an elliptic boundary, the nodal diameter would be fixed, viz. it would coincide with either the major or the minor axis, and the frequencies would be unequal. * See Bayleigh's treatise, Art. 339. A general formula for calculating the roots of J/ (£a) =0,. due to Prof. J. M«Mahon, is given by Gray and Mathews, p. 241. Numerioal tables are included in the collections of Dale, and Jahnke and Emde. 280 Tidal Waves [OHAP. vm 191] Properties of BesseCs Functions 281 The diagrams on the opposite page shew the contour-lines of the free surface in the first two modes of the present species. These lines meet the boundary at right angles, in conformity with the general boundary condition (Art. 190 (2)). The simple-harmonic vibrations of the individual particles take place in straight lines perpendicular to the contour-lines, by Art. 189 (4). The form of the sections of the free surface by planes through the axis of z is given by the curve y ^ J-i (x) on p. 278. The first of the two modes here figured has the longest period of all the normal types. In it, the water sways from side to side, much as in the slowest mode of a canal closed at both ends (Art. 178). In the second mode there is a nodal circle, whose radius is given by the lowest root of /^ (kr) = ; this makes r = •719a*. A oomparison of the preoeding investigation with the general theory of small oscilla- tions referred to in Art. 168 leads to several important properties of BesseVs Functions. In the first place, since the total mass of water is unaltered, we must have Ciir fa j I Crd0dr=O, (16) where ( has any one of the forms given by (4). For «>0 this is satisfied in virtue of the trigonometrical factor cos s3 or sin sB; in the symmetrical case it gives /: ''jo{kr)rdr=0 (17) Again, since the most general free motion of the system can be obtained by superposi- tion of the normal modes, each with an arbitrary amplitude and epoch, it follows that any value whatever of {, which is subject to the condition (16), can be expanded in a series of the form f =22 {Ag cos 80 +B, sin sB) J, (kr) (18) where the summations embrace all integral values of a (including 0) and, for each value of s, all the roots k of (5). If the coefficients Ag, B« be regarded as functions of t, the equa- tion (18) may be regarded as giving the value of the surface-elevation at any instant. The quantities Ag, B, are then the normal co-ordinates of the present system (Art. 168) ; and in terms of them the formulae for the kinetic and potential energies must reduce to sums of squares. Taking, for example, the potential energy V =igp jjCdxdy, (19) f2w fa this requires that I f w^w^rdBdr^O^ J J (20) * The oacillatione of a liquid in a circular basin of any uniform depth were disouased by Poisson, "Sur lea petites oscillations de Teau contenue dans un cylindre/' Ann, de OergonnCf t. xix. p. 225 (1828-9); the theory of Bessers FunctionB had not at that date been worked out, and the results were consequently not interpreted. The full solution of the problem, with numerical details, was given independently by Rayleigh, Phil Mag. (5), t. L p. 257 (1876) [Papers, t. L p. 25]. The investigation in the text is limited, of course, to the case of a depth small in oomparison with the radius a. Poisson's and Bayleigh's solution for the case of finite depth will be noticed in Chapter ix. 282 Tidal Waves [chap, vm where w^ , w^ are any two tenns of the expansion ( 18). UtOi.Wi involve coeines or sines of different multiples of 6, this is verified at once by integration with respect to B\ but if we take w^ oc t/« (hi r) cos sOf w^ x J. (^^'^) ^^^ '^» where hi, k^ are any two distinct roots of (5), we get /: a J. {k^r) J, (4,r) rrfr =0. (21) The general results, of which (17) and (21) are particular cases, are {''j^(kr)rdr=-^J^'(ka) (22) (cf. Art. 102 (10)), and /•« 1 In the case of ki =^, the latter expression becomes indeterminate; the evaluation in the usual manner gives I* {J. (te)}« rdr =-~ [ik«a« {J/ (ka)}^ + (fc«a« - «») {J. (ka)}*] (24) For the analytical proofs of these formulae we refer to the treatises cited on p. 129. The small oscillations of an annular sheet of water bounded by concentric circles are easily treated, theoretically, with the help of Bessel's Functions * of the second kind.' The only case of any special interest, however, is when the two radii are nearly equal ; we then have practically a re-entrant canal, and the solution follows more simply by the method of Art. 178. The analysis can also be applied to the case of a circular sector of any angle*, or to a sheet of water bounded by two concentric circular arcs and two radii. 192. As an example oi forced vibrations, let us suppose that the dis- turbing forces are such that the equilibrium elevation would be f=«©' cos 80 . COS (at + e) (25) This makes V^af = 0, so that the equation (12) of Art. 189 reduces to the form (1), above, and the solution is C = AJ, (*r) cos »e . cos (of + €), (26) where A is an arbitrary constant. The boundary-condition (Art. 189 (14)) gives AkaJ/ (ka) = sC, whence £ = C ^^^ ^ cos sB .cob (at + €) (27) The case « = 1 is interesting as corresponding to a uniform horizontal force; and the result may" be compared with that of Art. 179. ♦ See Rayleigh, Theory of Sound, Art. 339. 191-193] Basin of Variable Depth 283 From the case 8 = 2 we could obtain a rough representation of the semi- diurnal tide in a polar basin bounded by a small circle of latitude, except that the rotation of the earth is not as yet taken into account. We notice that the expression for the amplitude of oscillation becomes infinite when J/ (ka) = 0. This is in accordance with a general principle, of which we have already had several examples ; the period of the disturbing force being now equal to that of one of the free modes investigated in the preceding Art. 193*. When the sheet of water is of variable depth, the calculation at the beginning of Art. 189 gives, as the equation of continuity, dt" dx dy ^' The dynamical equations (Art. 189 (2)) are of course unaltered. Hence, eliminating ^, we find, for the free oscillations, ar^ = Haira^)^a^ravl ^'^ If the time-factor be «*<'<+•>, we obtain When h is a function of r, the distance from the origin, only, this may be written dhdC , <T« dr dr g l^^.K + T^ + ^-i-^ (4) As a simple example we may take the case of a oiroular basin which shelves gradually from the centre to the edge, according to the law =*.(!-.)• (6) Inteoduoing polar oo-ordinates, and assaming that ( yariee as cos »6 or sin a6, the equation (4) takee the form (^-a«,Ka;S+;:3^-^f)-J,'^+^f=0. (6) That integral of this equation which is finite at the origin is easily found in the form of an ascending series. Thus, assuming f'^^"©" (') where the trigonometrical factors are omitted, for shortness, the relation between consecu- tive coefficients is found to be (m«-««)^^ = |m(m-2)-^-^1^,,.„ * This formed Art. 189 of the 2nd ed. of this work (1896). A similar investigation is given by Poincar6, LeQona de mdcanique eOtaU, t. iil. (Paris, 1910). 284 Tidal Waves [chap, vm o-»a« or, if we write -j— =n (n - 2) - «■, (8) where n is not as yet assumed to be integral, (m«-*«)24,„=(w-n)(m+»-2)24^., (9) The equation is therefore satisfied by a series of the form (7), beginning with the term At (r/ay, the succeeding coefficients being determined by putting m=«+2, «+4, ... in (0). We thus find A^A /''"V/i (n-s-2){n+8) f^ . (n-a-4)(n-^-2)(n+g)(n-fj+2) r* \ .^ ^ 'W t 2(2«+2) ^^ 2. 4 (2« +2) (2^+4) a^-"'j'^'''> or in the usual notation of hypergeometric series f=^J..j(«./3,y.5) (11) where 0=^71+^8, /3 = 1+J«— Jn, y=« + l. Since these make y -a -/3=0, the series is not convergent for r=a, unless it terminate. This can only happen when n is integral, of the form a +2j. The corresponding values of <r are then given by (8). In the symmetrical modes (« =0) we have where j may be any integer greater than unity. It may be shewn that this expression vanishes for J —1 values of r between and a, indicating the existence oij—l nodal circles. The value of <r is given by <r«=4j(i-l)''^ (13) Thus the gravest symmetrical mode {j=2) has a nodal circle of radius -TOTa; and its frequency is determined by <r" = 8^Ao/^*' Of the unsymmetrical modes, the slowest, for any given value of 8, is that for which n =s« +2, in which case we have r* f =-4, — cos *tf cos (erf + €), the value of <r being given by o^=28 , ^ (14) The slowest mode of all is that for which 8 = 1, n =3; the free surface is then alwa3r8 plane. It is found on comparison with Art. 191 (15) that the frequency is *768 of that of the corresponding mode in a circular basin of uniform depth h^, and of the same radius. As in Art. 192 we could at once write down the formula for the tidal motion produced by a uniform horizontal periodic force ; or, more generally, for the case where the disturbing potential is of the type Q ccr* cos 8$ cos {ai + c). 194. We may conclude this discussion of Uong' waves on plane sheets of water by an examination of the mode of propagation of disturbances from a centre in an unlimited sheet of imiform depth. For simplicity, we will consider only the case of symmetry, where the elevation ^ is a function of 193-194] BesseVs Function of the Second Kind 285 the distance r from the origin of disturbance. This will introduce us to some peculiar and rather important features which attend wave-propagation in two dimensions. The investigation of a periodic disturbance involves the use of a Bessel's Function (of zero order) ' of the second kind,' as to which some preliminary notes may be useful. To solve the equation ^ '^~ d "^^'^ ^^^ by definite integrals, we assume* == f ^~'' ^^» (2) where 2^ is a function of the complex variable t, and the limits of integration are constants as yet unspecified. This makes ^dz* by a partial integration. The equation (1) is accordingly satisfied by *=/ir;««l ('> provided the expression V(l+(")e-* vanish at each Umit of integration. Hence, on the supposition that z is real and positive, or at all events has its real part positive, the integral in (3) may be taken along a path joining any two of the points i, - 1, + oo in the plane of the variable t ; but two distinct paths joining the same points will not necessarily give the same result if they include between them one of the branch-points (t = ±:i) of the function under the integral sign. Thus, for example, we have the solution *^ = A >/(i+tr where the path is the portion of the imaginary axis which lies between the limits, and that value of the radical is taken which becomes = 1 for ^ =0. If we write ^ =£ +^17, we obtain <^ = »r J^J^j=2tJJ%08(2COS^)(W=»,rJo(2) (4) which is the solution already known (Art. 100). An independent solution is obtained if we take the integral (3) along the axis of 17 from the point (0, i) to the origin, and thence along the axis of £ to the point (00 > 0). This gives, with the same determination of the radical, fO e-*'^d(irf) r e-'U( _^ r e-'^dj _. f^e-''^dfj *""j< v/(l-'7») '^Josf{l+P)~)os^(l+$') *ioV('l-»7») By adopting other pairs of limits, and other paths, we can obtain other forms of <p, but these must all be equivalent to <f>i or <^2* ^^ ^ linear combinations of these. In particular, some other forms of 02 ^^ important. It is known that the value of the integral (3) taken * Forsyth, Differeniial Equations, c. vii. The systematic application of this method to the theory of BesseFs Functions is due to Hankel, "Die Cylinderfunktionen erster u. zweiter Art," Maik, Ann. t. i p. 467 (1869). See Gray and Mathews, 0. vii. (6) 286 Tidal Waves [chap, vm round any closed contour which excludes the branch-points (t = ±^i) is zero. Let us first take as our contour a rectangle, two of whose sides coincide with the positive portions of the axes of £ and 7, except for a small semicircular indentation about the point t =t, whilst the remaining sides are at infinity. It is easily seen that the parts of the integral due to the infinitely distant sides wiU vanish, either through the vanishing of the factor c~*^ when f is infinite, or through the infinitely rapid fluctuation i^- of the fimction e'^'^/rj when 17 is infinite. Hence for the path which gave us (5) we may substitute that which extends along the axis of 17 from the point (0, i) to (0, too ), provided the continuity of the radical be attended to. Now as the variable t travels counter- ^ clockwise round the small semicircle, the radical changes continuously from V(l— »?*) *<> *V(»7*— !)• We have therefore (6) It will appear that this solution is the one which is specially appropriate to the case of diverging waves. Another method of obtaining it will be given in Chapter x. If we equate the imaginary parts of (5) and (6) we obtain a form due to Mehler*. 2 r *^o (*) =- I si'i (^ cosh u) du, (7) On account of the physical importance of the solution (6) it is convenient to have a special notation for it. We writef Do(z)=- r«-<««»^»du (8) "■ y This is equivalent to Dq (z) =Zo (2) -tJ© (*)» W 2 r wherej K^^ (z) =- I cos (z cosh u)du (10) ^ y Equating the real parts of (5) and (6) we have, also, ^0(2)=^- r «-"»"*» "rfi*-? f *' sin (z cos 5) (is (11) IF J »r y ♦ Maih, Ann. t. v. (1872). t 2>o (z) is equivalent to - iH^^ (z) in the notation proposed by Nielsen. X As regards K^, this is the notation employed by Heine (except as to the constant factor), and H. Weber. The reader should be warned, however, that the same symbol has been employed in at least two other distinct senses in connection with the theory of BessePs Functions. From a purely mathematical point of view the choice of a standard solution *of the second kind' is largely a matter of convention, since the differential equation (1) is etill satisfied if we add any constant multiple of /q (z). In terms of the notation introduced by C. Neumann, which has found some acceptance, K, i") =1{-Yt (?) + (log 2 - y) J, (»)}. IT where 7 = '6772. . . (Enler*8 constant). A table of the function iirKQ (z) has been constructed by B. A. Smith ; see PhU. Mag. (5), t.xlv. p. 122(1898). 194] BesseHs Function of the Second Kind 287 For a like reason, the path adopted for ^^ may be replaced by the line drawn from the point (0» i) parallel to the axis of f (viz. the dotted line in the figure). To secure the con- tinuity of ^(\ -k-t^), we note that as i describes the lower quadrant of the small semicircle, the value of the radical changes from J(\ - 17') to e^**^ V(2£)> approximately. Hence along the dotted line we have, putting /=»+£, where that value of the radical is to be chosen which is real and positive when f is infini- tesimal Thus If we expand the binomial, and integrate term by term, we find ^.,.,.(l)V..-*j..|^(-).!^(i)'....) ,.3, where use has been made of the formulae If we separate the real and imaginary parts of (13) we have, on comparison with (9), Jo (2) = (;^)* {5 sin (« + l»r) - >8f cos (2 + ifr) }, (16) Ko(z)={^^ {BQo%(z+\'!T)^89m(z+\fr)}, (16) / (14) (17) , _ , 1«.3« 12.3*.6«.7* ^*'^"' ^ = ^""2T(8i?"*' 4!(8z)* •• 1« 1^3'. 5^ ^"17(8^" 3!(8z)» """' The series in (13) and (17) are of the kind known as * semi-convergent,* or 'asymptotic,' expansions ; i.e. although for sufficiently large values of z the successive terms may for a while diminish, they ultimately increase again indefinitely, but if we stop at a small term we get an approximately correct result*. This may be established by an examination of the remainder after m terms in the process of evaluation of (12). It follows from (16) that the large roots of the equation Jy (z) =0 approximate to those of sm (z + Jir) -0 (18) The series in (13) gives ample information as to the demeanour of the function Do (z) when z is large. When z is small, D© (z) is very great, as appears from (9) and (11). An approximate formula for this case can be obtained as follows. Referring to (11), we have /;,— ..../%-K)^./;t±!{..i.'(i)-....)* ■ll'^{'*s,*h{£}*-]*'- "?> * Gf. Whittaker, Modem Analysis^ c. viii. ; Bromwich, Theory of Infinite Series^ London 1908, 0. xL The semi-convergent expansion of J^ (z) is due to Poisson, Joum, de V£c6U Polyt, cah. 19, p. 349 (1823) ; a rigorous investigation of this and other analogous expansions was given by Stokes, l.c, ante p. 277. The * remainder' was examined by lipschitz, CreUe, t. Ivi. p. 189 (1869). Of. Hankel, Ic^ ante p. 286. 288 Tidal Waves [chap, vin The first term gives* d«^=-y-logJz + ..., (20) and the remaining ones are small in comparison. Hence, by (9) and (11), ^o(2)=--(logi2 + y + itV + ...) (21) IT 2 It follows that lim zDq (2) = - - f (22) 8=0 ^ The formula (21) is sufficient for our purposes, but the complete expression can now be obtained by comparison with the general solution of (1) in terms of ascending series, viz. % Jo(2)logr + 22"*«2n«"^*«2n«76«"**7 ^^^^ 1 1 1 2"*"3'^*""^m' where «m = l +s + « + ...+- • In order to identify this with (21), for small values of z, we must make 2 2 5= , ^=--(logi+y+i»ir) (24) TT IT Hence 2 2 fz' z^ z* 1 ^o(g)=~-(iogig+y+^*«')«^o(»)~- |22^"^«2^1«^'^^» 2^4^6« ~'•T (26) 195. We can now proceed to the wave-problem stated at the beginning of Art. 194. For definiteness we will imagine the disturbance to be caused by a variable pressure p^ applied to the surface. On this supposition the dynamical equations near the beginning of Art. 189 are replaced by dt ^dx pdx' dt pdy p dy ' ^' if=-»(i+i) ,2, as before. If we introduce the velocity-potential in (1), we have, on integration, I-K + & (3) We may suppose that Pq refers to the change of pressure, and that the arbi- trary function of t which has been incorporated in <f) is chosen so that d<f>ldt = * De Morgan, Differential and Integral Calculus, London, 1842, p. 653. t The Beesers Functions of the second kind were first thoroughly investigated and made ayailable for the solution of physical problems in an arithmetically intelligible form by Stokes, in a series of papers published in the Camb. Trans. With the help of the modem Theory of Functions, some of the processes have been simplified by Lipschitz and others, and (especially from the physical point of view) by Rayleigh. These later methods have been freely used in the text. X Gray and Mathews, p. 11. 194-195] Waves Diverging from a Centre 289 in the regions not afiected by the disturbance. Eliminating { by means of (2), we have |^.^v.v + i|" W When ^ has been determined, the value of ^ is given by (3). We will now assume that f^ is sensible only over a small* area about the origin. If we multiply both sides of (4) by 8x8^, and integrate over the area in question, the term on the left-hand may be neglected (relatively), and we find 'Pi^-jphilh^^y^ (^) gph dt, where hs is an element of the boundary of the area, and Sn refers to the hori- zontal normal to hs, drawn outwards. Hence the origin may be regarded as a two-dimensional source, of strength /("-^-ijt ■ <«) where Pq is the integral disturbing pressure. Turning to polar co-ordinates, we have to satisfy where c* = gh, subject to the condition ,"^(-^l)=-^(^> («) where/ (i) is the strength of the source, as above defined. In the case of a simple-harmonic source e*'* the equation (7) takes the form '^ + i| + *V-o. (9) where k = a/c, and a solution is .^ = iZ)o(AT)6-*, (10) where the constant factor has been determined by Art. 194 (22). Taking the real part we have <f> = i {Kq (kr) cos at + Jq (^) sin aQ, (11) « corresponding to / (0 = ^^ ^^' * That is, the dimennons of the area are small compared with the * length' of the waves generated, this term being understood in the general sense of Art 172. On the other hand, the dimensions must be supposed large in compcuison with h» L.H. 19 2»0 Tidal Waves [chap, vm For large values of hr the result (10) takes the form The combination t — rfc indicates that we have, in fact, obtained the solution appropriate to the representation of diverging waves. It appears that the amplitude of the annular waves ultimatdy varies inversely as the square root of the distance from the origin. 196. The solution we have obtained for the case of a simple-harmonic source c*'* may be written •00 iff(t-- ooflh u ] 2nr<f> = j e ' du (13) This suggests generalization by Fourier's Theorem ; thus the formula 2^(f, = j fU^^coahujdu (U) should represent the disturbance due to a source /(^ at the origin*. It is implied that the form oif(t) must be such that the integral is convergent; this condition will as a matter of course be fulfilled whenever the source has been in action only for a finite time. A more complete formula, embracing both converging and diverging waves, is 2nr<f)=j f(t--coBhu\du+j F (t + -cosh in du (15) The solution (15) may be verified, subject to certain conditions, by substitution in the differential equation (7). Taking the first term alone, we find = I •Isinh' u.f" (t — cosh tt J — cosh u.f'U — cosh « U dw This obviously vanishes whenever /(<)=0 for negative values of t exceeding a certain limitf. * The sahstanoe of Arts. 196 — 198 is adapted from a paper "On Wave-Propagation in Two Dimensions,*' Proc, Lond. Math. 8oc, t. xxxv. p. 141 (1902). A result equivalent to (14) was obtained (in a different manner) by Levi-Civita, Nuopo dmerUo (4), t. vi. (1897). t The verification is very similar to that given by Levi-Civita. 1 95-1 97] Waves Diverging from a Centre 291 Again, -2irr^=^/ ooBh«./M< — -coshttMu = ^| (8inhit+c-«)/' (<-%oshtt^dtt under the same condition. The limiting value .of this when r-»-0 is /(<); &nd the state- ment made above as to the strength of the source in (14) is accordingly verified. A similar process will apply to the second term of (15) provided F (t) vanishes for positive values of t exceeding a certain limit-. 197. We may apply (14) to trace the effect of a temporary source varying according to some simple prescribed law. If we suppose that everything is quiescent until the instant t = 0, so that f{t) vanishes for negative values of i, we see from (14) or from the equivalent form 27r^.= | J-^^l^ (16) that <f> will be zero everywhere so long as i < r/c. If, moreover, the source acts only for a finite time r, so that/ (t) = for i > r, we have, for <> r + r/c, P_0fli9 ,„)• This expression does not as a rule vanish ; the wave accordingly is not sharply defined in the rear, as it is in front, but has, on the contrary, a sort of 'tail''|' whose form, when < — r/c is large compared with r, is determined by 27rA 1 ifVW<W (18) (<«*-f«/c*)' The elevation { at any point is given by (3), viz. £-^|-- m It follows that / 00 Cdt = 0, (20) -00 * Analytically, it may be noticed that the equation (4), when Po=^> °^7 ^ written and that (17) consiBtB of an aggregate of solutions of the known type t The existence of the 'tail' in the case of oylindrical electric waves was noted by Heaviside, PhU. Mag, (6), t. xxvi (1888) [Electrical Papers, t. ii], 19—2 292 Tidal Waves [chap, vm provided the initial and final values of ff> vanish. It may be shewn that this will be the case whenever /(f) is finite and the integral / oo /(O* (21) — 00 is convergent. The meaning of these conditions appears from (6). It follows that even when P© is always positive, so that the flux of liquid in the neighbourhood of the origin is altogether outwards, the wave which passes any point does not consist sblely of an elevation (as it would in the corresponding one-dimensional problem) but, in the simplest case, of an elevation followed by a depression. To illustrate the progress of a solitary wave we may assume /W = ?T^' (22) which makes P^ increase from one constant value to another according to the law Po = il + 5 tan-i - ( 23 ) The disturbing pressure has now no definite epoch of beginniiig or ending, but the range of time within which it is sensible can be made as small as we please by diminishing r. For purposes of calculation it is convenient to assume /W = ,-^ (24) in place of (22), and to retain in the end only the imaginary part. We have then 2,.^=r_;^-*i — =2r-;^ — TT-^r ^^^ I t — coshtt-tV I t IT— (< + — ir\z^ Jo c Jo c \ c J I where z =:tanh ^u. We now write ^_!:-»V=a«e-«^ t+--tr=6«e-«^ (26) c c where we may suppose that a, h are positive, and that the angles a, /3 lie between and \ir» Since )■ (27) tan2a=— — , tan 2/3 = -: — » it appeahf that'ct ^ !>*according as f ^ 0, and that a > /3 always. With this notation, we find 2-^C To interpret the logarithms, let us mark, in the plane of a complex variable z, the points. 197] SolUary Wave 293 Since the integral in the seoond member of (28) is to be taken along the path 01^ the proper value of the third member is ((logg + t.OP/)-(logg-i.OG/)}, a where real logarithms and positive values of the angles are to be understood. Hence, rejecting all but the imaginary part, we find a . sin(a+/3), IP co8(a+/3), «,^, ,^, 2^»= ab ^ ^gjQ^ ^;r~^(^-^^0) (29) as the solution corresponding to a source of the type (22). Here IP _ /a*+2ab cos (a +/3) +6«\i 2a6sin(a~ffl /0"V-2a6c6s(7->)+6V ' **^^^^- grZ^i (^) and the values of a, d, a» 3 in terms of r and t are to be found from (27). It will be sufficient to trace the effect of the most important part of the wave as it passes a point whose distance r from the origin is large compared with or. If we confine ourselves to times at which i —r/c is small compared with r/e, a will be small compared with 6y PIQ will be a small angle, and IP/IQ will = 1, nearly. If we put < = -+Ttani;, (31) we shall have a=lir-i^» a = V(rseci7), /3=W» &=(2r/c)*, (32) approximately; and the formula (29) will reduce to 2jr<^=^cosa=-~f^j cos (Jir - Ji;) V(C08 >?) (33) 294 Tidal Waves [ohap- vm The elevation C la then given by approximately. The diagram on the preceding page shews the relation between ( and i^ as given by this formula*. 198. We proceed to consider the case of a spherical sheet, or ocean, of water, covering a solid globe. We will suppose for the present that the globe does not rotate, and we will also in the first instance neglect the mutual attraction of the particles of the water. The mathematical conditions of the question are then exactly the same as in the acoustical problem of the vibrations of spherical layers of airf . Let a be the radius of the globe, h the depth of the fluid; we assume that h is small compared with a, but not (as yet) that it is imiform. The position of any point on the sheet being specified by the angular co-ordinates Oy ff>y. Jet. K .b^ the component velocity of the fluid at this point along the meridian, in the direction of 9 increasing, and v the component along the parallel of latitude, in the direction of ff> increasing. Also let ^ denote the elevation of the free surface above the nndisturbed level. The horizontal * ■ « motion being assumed, for the reasons explained in Art. 172, to be the same at all points in' a vertical line, the condition of continuity is ^(wAasine&^)8& + ^(vAa8fl)8^ = ~a8ine&^.a8e.|^, where the left-hand side measures the flux out of the columnar space standing on the element of area a sin dh<f} . aSd, whilst the right-hand member expresses the rate of diminution of the volume of the contained fluid, owing to fall of the surface. Hence dt _ 1 ja (hu sin e) d (hv)\ f (1) dt a sin fl i dd ^ d<f> If we neglect terms of the second order in t^, t;, the dynamical equations are, on the same principles as in Arts. 169, 189, dt ^ add add' dt ^asinflg^ asin^a^' • ' ' '^ ' where Ci denotes the potential of the extraneous forces. If we put f = - Q/flf, (3) these may be written ?^-._._«9^ „ 3^ asin^a^;^ ^^' ......W * The pointe marked - 1, 0« + 1 on the diagram correspond to the times r/c-T, rfCf rjc +r, respectively. t IMsoiissed in Rayleigh's Theory of Sound, c. zviii 197-199] Wave% on a Spherical Sheet 29& Between (1) and (4) we can eluninate u, v, and bo obtain an equation in £ Only. In the case of simple-hannonic motion, the time-factor bdng e*<'*+•^ the equations take the forms y i j 9 (fusing) 8 {hv) ) .^. ^""oasinflt dd ^ d<l> r '"^ ^ 199. We will now consider more particularly the case of uniform depth. To find the free oscillations we put ^ = ; the equations (5) and (6) of the preceding Art. then lead to This is identical in form with the general equation of spherical surface^ harmonics (Art. 83 (2)). Hence, if we put a«a« , = w (n + 1), (2) a solution of (1) will be C = ^n> (3) where 8^ is the general surface-harmonic of order n. It was pointed out in Art. 86 that 8^ will not be finite over the whole sphere unless n be integral. Hence, for an ocean covering the whole globe, the form of the free surface at any instant is, in any fimdamental mode, that of a 'harmonic spheroid' r = a + A + iSn cos (<7« + €), (4) and the speed of the oscillation is given by a = {n(n+l)}*.^, (5) the value of n being integral. The characters of the various normal modes are best gathered from a study of the nodal lines (S„ = 0) of the free surface. Thus, it is shewn in treatises on Spherical Harmonics* that the zonal harmonic P„ (/n) vanishes for n real and distinct values of /x lying between ± 1, so that in this case we have n nodal circles of latitude. When n is odd one of these coincides with the equator. In the case of the tesseral harmonic a-'")"'^'rW. * For refeienceB see p. 103. 20ft Tidal Waves [chap, vin the second factor vanishes for n — 8 values of /i, and the trigonometrical factor for 2a equidistant values of <f>. The nodal lines therefore consist of n — 8 parallels of latitude and 28 meridians. Similarly the sectorial harmonic has as nodal lines 2n meridians. These are, however, merely special cases, for since there are 2n 4- 1 independent surface-harmonics of any integral order n, and since the frequency, determined by (5), is the same for each of these, there is a corresponding degree of indeterminateness in the normal modes, and in the configuration of the nodal lines. We can also, by superposition, build up various types of progressive waves; e.g. taking a sectorial harmonic we get a solution in which f « (1 - /x*)*** cos (tk^ - <rf + €) ; (6) this gives a series of meridianal ridges and furrows travelling round the globe, the velocity of propagation, as measured at the equator, being ?-("-^)'-w* <') It 18 easily verified, on examination, that the orbits of the particles are now ellipses having their principal axes in the directions of the meridians and parallels, respectively. At the equator these ellipses reduce to straight lines. In the case n = 1, the harmonic is always of the zonal type. The harmonic spheroid (4) is then, to our order of approximation, a sphere excentric to the globe. It is important to remark, however, that this case is, strictly speaking, not included in our djmamical investigation, unless we imagine a constraint applied to the globe to keep it at rest; for the de- formation in question of the free surface would involve a displacement of the centre of mass of the ocean, and a consequent reaction on the globe. A corrected theory for the case where the globe is free could easily be investigated, but the matter is hardly important, first because in such a case as that of the Earth the inertia of the solid globe is so enormous compared with that of the ocean, and secondly because disturbing forces which can give rise to a deformation of the type in question do not as a rule present themselves in nature. It appears, for example, that the first term in the expression for the tide-generating potential of the sun or moon is a spherical harmonic of the 8ec<md order (see the Appendix to this Chapter). When n = 2, the free surface at any instant is approximately ellipsoidal. The corresponding period, as found from (5), is then -816 of that belonging to the analogous mode in an equatorial canal (Art. 181). 199-^200] Free and Forced OmUations 297 For large values of n the distance from one nodal line to another is small compared with the radius of the globe, and the oscillations then take place muchas OH a plane sheet of water. For example, the velocity of propagation, at the equator, of the sectorial waves represented by (6) tends with increasing n to the value (jrA)*, in agreement with Art. 170. From a compariiBOn of the foregoing investigation with the general theory of Art. 168 we are led to infer, on physical grounds alone, the possibility of the expansion of any arbitrary value of f in a series of surface harmonics, thus 00 the coefficients of the various independent harmonics being the normal co-ordinates of the system. Again, since the products of these coefficients must disappear from the expressions for the kinetic and potential energies, we are led to the * conjugate' properties of spherical harmonics quoted in Art. 87. The actual calculation of the energies will be given in the next Chapter, in connection with an independent treatment of the same problem. The ejBEect of a simple-harmonic disturbing force can be written down at once from the formula (14) of Art. 168. If the surface value of Q be expanded in the form Q = SQ„, (8) where Q^ is a surface-harmonic of integral order n, the various terms are normal components of force, in the generalized sense of Art. 135; and the equilibrium value of f corresponding to any one term ii„ is Hence, for the forced oscillation due to this term, we have ^• — T^'T' -^ where a measures the * speed' of the disturbing force, and a^ that of the corresponding free oscillation, as given by (5). There is no difficulty, of course, in deducing (10) directly from the equations of the preceding Art. 200. We have up to this point neglected the mutual attraction of the parts of the liquid. In the case of an ocean covering the globe, and with such relations of density as we meet with in the actual earth and ocean, this is not insensible. To investigate its effect in the case of the free oscillations, we have only to substitute for Q„, in the last formula, the gravitation- potential of the displaced water. If the density of this be denoted by p, whilst po represents the mean density of the globe and liquid combined, we have* ".=-£nf <") and 9 = iY^(^Po (12) * See, for example, Bouth, Analytical Statics, 2nd ed., Cambridge, 1902, t. ii pp. 146-7. 298 Tidal Waves [qHAP, vm y denoting the gravitation-constant, whence "-=-2;nn[-^^^- -(i^i Substituting in (10) we find ^^('-i^^ ("' where a^ is now used to denote the actual speed of the oscillation, and u^ the speed calculated on the former hypothesis of no mutual attraction. Hence the corrected speed is given by '■■-<"+')('-2;rTi£)S ('»' 2n + 1 pj a^' For an ellipsoidal oscillation (n = 2), and for p/po = '18 (as in the case of the Earth), we find from (14) that the effect of the mutual attraction is to lotoer the frequency in the ratio of '94 to 1. Hie slowest oscillation would correspond to n = 1, but, as already indicated, it would be necessary, in this mode, to imagine a constraint applied to the globe to keep it at rest. This being premised, it appears from (15) that if p> Po the value of ci* is negative. The circular function of f is then replaced by real exponentials ; this shews that the configuration in which the surface of the sea is a sphere concentric with the globe is one of unstable equilibrium. Since the introduction of a constraint tends in the direction of stability, we infer that when p > po the equilibrium is a fortiori unstable when the globe is free. In the extreme case where the globe itself is supposed to have no gravitative power at all, it is obvious that the water, if disturbed, would tend ultimately, under the influence of dissipative forces, to collect itself into a spherical mass, the nucleus being expelled. It is obvious from Art. 168, or it may easily be verified independently, that the forced vibrations due to a given periodic disturbing force, when the gravitation of the water is taken into account, will be given by the formula (10), provided ii„ now denote the potential of the extraneous forces only, and cr„ have the value given by (15). 201. The oscillations of a sea bounded by meridians, or parallels of latitude, or both, can also be treated by the same method f. The spherical harmonics involved are however, as a rule, no longer of integral order, and it is accordingly difficult to deduce numerical results. * This reanlt was given by Laplace, Micaniqut CAeste, livre l*", Art. 1 (1799). The tree and the forced oscillations of the tjrpe n=2 had been previously investigated in his "Becherohes snr quelques points du syst^me du monde/* M4m. de VAcad, roy, des ScienceSf 1775 [1778] [Oeuvre* Computes, t. ix. pp. 109, . . .]. t Of. Rayleigh, U. anU p. 294. . 200^201] Waves on a Limited Ocean 299 In the case of a zonal sea bounded by two parallels of latitude, we assnme C={ApW+Bq{^)}^js<l, (1) where fi^ooeS, and p (/i), q (fi) are the two fonctions of /t, containing (1 — /i*)^ as a factor, which are given by the formula (2) of Art. 86. It will be noticed that |> (/i) is an even, and q (fi) an odd function of /a. If we distinguish the limiting parallels by suffixes, the boundary conditions are that u =0 ioT fj, =s fi^ and fJL=fif For the free oscillations this gives, by Art. 198 (6), 4p'(Mi)+5^(/*i)=0. ^p'(/*a)+5/(M«)=0 (2) whence =0. (3) P' (Ht)f ^' (h) which is the equation to determine the admissible values of n, the order of the harmonics. The speeds (o-) corresponding to the various roots are given as before by Art. 199 (5). If the two boundaries are equidistant from the equator, we have /x^ = —fi^ • The above solutions then break up into two groups ; viz. for one of these we have 5=0, i>'(/ii)=0, (4) and for the other -4=0, 3^(fii)=0 (6) In the former case C has the same value at two points symmetrically situated on opposite sides of the equator; in the latter the values at these points are numerically equal, but opposite in sign. If we imagiue one of the boundaries to be contracted to a point (say /i^ = 1), we pass to the case of a circular basin. The values of ^' (1) and 9' (1) are infinite, but their ratio can be evaluated by means of formulae given in Art. 84. This gives, by the second of equations (2), the ratio A : B, and substituting in the first we get the equation to determine n. A simpler method of treating this case consists, however, in starting with a solution which is known to be finite, whatever the value of n, at the pole /i = l. This involves a change of variable, as to which there is some latitude of choice. We might take, for instance, the expression for P^* (cos 6) in Art. 86 (6), and seek to determine n from the condition that for^ = ^i*. ^P,-(coe<?)=0 (6) By making the radius of the sphere infinite, we can pass to the plane problem of Art. 191 1« The steps of the transition will be understood from Art. 100. If the sheet of water considered have as boundaries two meridians (with or without parallels of latitude), say ^=0 and <^ =a, the condition that v=0 at these restricts us to the factor cos sa, and gives 8a =mir, where m is integral This determines the admissible values of «, which are not in general integral :(. * This question has been discussed by Maodonald, Proc Lond. Math. 8oe. t. xxzi. p. 264 (1899). t Cf. Bayleigh, Theory of Sound, Arts. 336, 338. X The reader who wishes to carry the study of the problem farther in this direction is referred to Thomson and Tait, Nakiral PhUoiojJiy (2nd ed.). Appendix B, "Spherical Harmonic Analysis." 300 Tidal Waves [chap* vm Tidal OsdUations of a Rotating Sheet of Water. 202. The theory of the tides on an open sheet of water is seriously complicated by the fact of the earth's rotation. If, indeed, we could assume that the periods of the free oscillations, and of the disturbing forces, were small compared with a day, the preceding investigations would apply as a first approximation, but these conditions are far from being fulfilled in the actual circumstances of the earth. The difficulties which arise when we attempt to take the rotation into account have their origin in this, that a particle having a motion in latitude tends to keep its angular momentum about the earth's axis unchanged, and so to alter its motion in longitude. This point is of course familiar in connection with Hadley's theory of the trade- winds*. Its bearing on tidal theory seems to have been first recognised by Maclaurint. Owing to the enormous inertia of the solid body of the earth compared with that of the ocean, the ejSect of tidal reactions in producing periodic changes of the angular velocity is quite insensible. This angular velocity will therefore for the present be treated as constant :|:. The theory of the small oscillations of a dynamical system about a state of equilibrium relative to a solid body which rotates with constant angular velocity about a fixed axis difEers in some important particulars from the theory of small oscillations about a state of absolute equilibrium, of which some account was given in Art. 168. It is therefore worth while to devote a little space to it before entering on the consideration of special problems. 203.' Let us take rectangular axes x, y, z fixed relatively to the solid, of which the axis of z coincides with the axis of rotation, and let <o be the angular velocity of the rotation. The equations of motion of a particle m relative to these moving axes are known to be w (^ - 2coy - w^x) = Z, m(y + 2ct}x- w^y) = 7, mz = Z, . .(1) where X, Y, Z are the impressed forces. From these we derive Swi {so Ax + yAy + i?Az) -f 2aiEm (xAy — yAx) - co»Sm (xAx -h yAy) = 2 {XAx + YAy + ZAz), . .(2) where the symbol A has the same meaning as in Art. 135. Let us now suppose that the relative co-ordinates (x, y, z) of each particle ♦ "The Cause of the General Trade Winds," Phil. Trans, 1735. t l>e Causd Physied Fluxus et Rtftuxua Maris, Prop, yii : "Motus aqusa turbatnr ex inseqtiali velooitate qui corpora circa azem Terrne motu diomo defenintur** (1740). X The secviar effect of tidal friction in this respect will be noticed later (Chapter ix). 202-203] Dynamics of a Rotating System 301 can be expressed in terms of a certain number of independent quantities 9i9 9i> ••• 9n9 ^^^ ^^^ '^ write ® = iSm (A« + ya + i*), To = W^rn (x^ + y*) (3) Here ^ denotes the energy of the relative motion, which we shall suppose expressed as a homogeneous quadratic function of the generalized velocitiea 9,., with coefficients which are functions of the generalized co-ordinates q^\ whilst Tq denotes the energy of the system when rotating with the solid, without relative motion, in the configuration (g^, 9t, ... 9n)* As ^ ^be proof of Lagrange's equations (cf. Art. 135) we find V /-A ^"A . :,A X 1^^ ^^\k^^(^^^ ^\ A 2 whilst a>«Sm(a^Ax + yAy) = ^Agi + ^Ag.+ ..,+|£"Ag (5> Also 2ci>Sw (^Ay - y Aa?) = (j3n?i + jSjj j, + • - • + Pmin) A?i + (^tl?l + ^22?2 + . . • + ^2n?„) Agr, + where j3,, = 2coSm //^' y\ , (7) and it is particularly to be noticed that Finally, we put S (ZAa? + YAy + ZAz) = ~ ^V -\' Q^liq^ + Q^^q^ + . . . + QnAg„, . .(9) where V is the potential energy, and Qj, Qj* ••• Qn are the generahzed comi)onents of disturbing force. If we substitute from (4), (5), (6) and (9) in (2), and equate separately the coefficients of Ajj, Ag2, • • • Ag„, we obtain n equations of the type* It may be noticed that these equations may be obtained as a particular case of Art. 141 (24), with the help of Art. 142 (8), by supposing the rotating solid to be free, but to have an infinite moment of inertia. < * Cf. Thomson and Tait, Natural PhUoaophy (2nd ed.), Part i. p. 319. 802 Tidal Waves [chap, vra The conditions for relative equiUbrium, in the absence of disturbing forces, are found by putting ji, 92* • • • jn = ^ ii^ {10)> ^^ more simply from (2). In either way we obtain |-(F-!ro) = 0. (11) 1 which shews that the equilibrium value of the expression 7 — T© is 'stationary.' If T denote the total kinetic energy of the system, we have T = \i:m{(x - a>y)« + (y + o^xf + i«} = ® + To + ^Sm (xy - yx), . .(12) whence, on reference to (1), J(r4-7) = |(3r+ro+ F) + coSm(a;y-y^) = |(®+ F-ro) + ^s(a?y-y^) (13) This is to be equated to the rate at which the disturbing forces do work, i.e. to a>S (xY - yX) + Qi9i + Qjft + . . . + Qnin^ Hence | (® + ^ "^ ^0) = OWi + ^292 + • • • + (?n?n (14) This result may also be deduced from the equations (10). When there are no disturbing forces, we have 3r+ 7-^0 = const (15) It may be noticed that the analogue of the Hamiltonian equation (20) of Art. 135 is now f '[A{®+ ro + coSw(»y-yic)} + S(ZAx+ yAy + ZA2)]cft = 0, ..(16) with the condition that the variations Ax, Ay, Az of the rdaJtive co-ordinates must vanish at both limits. This follows easily from (1). If we transform to generalized co-ordinates, using (9), we may derive an independent proof of the equations (10). 204. We will now suppose the co-ordinates q^ to be chosen so as to vanish in the undisturbed state. In the case of a amaJl disturbance, we may then write 2® = Oii^i* + a^q^ + . . . + 2ai2?i?2 + (1) 2(7- To) " Ciig* + ^22?' + . . . + 2cx2gi?2 + • • . , (2) where the coefficients may be treated as constants. The terms of the first degree in 7 — T© ^*ve been omitted, on account of the 'stationary' property. 203-206] Principal Co-ordinates 303 In order to simplify the equations as much as possible^ we will further suppose that, hj a linear transformation, each of these expressions is reduced, as in Art. 168, to a sum of squares ; viz. 23r = ai?i" + a2?2" + . . - + Mn*, (3) , 2 (7 - To) - CiJi« + c^qj" + • . . + c^qn^ (4) The quantities 9i, jt, ... qn niay be called the 'principal co-ordinates' of the system, but we must be on our guard against assuming that the same simplicity of properties attaches to them as in the case of no rotation. The coefficients o^, a,, ... a„ and c^, Cg, ... c,, may be called the 'principal coefficients' of inertia and of stability, respectively. The latter coefficients are the same as if we were to ignore the rotation, and to introduce fictitious 'centrifugal' forces (nuo^x, mwh/, 0) acting on each particle in the direction outwards from the axis. The equations ^10) of the preceding Art. become, in the case of infinitely small motions, «i§i + ^i?i + Pnqt + ftsft + . . . -f Pmqn = Qly «2?i + cjja + Ptiqi + Puqz + . . . + Ptnqn = Qi, [ (5) «n?n+ <^nqn-^ ^nl?l + ^fil?2 + ^n8?8 + • • • = Qnf where the coefficients Prs ^^7 ^^ regarded as constants. If w:e multiply these hj qi, qt, ... ?» ^ order and add, we find, taking account of the relation j5„ = — ftr* |(®+l'-2'o) = ei?l + e2?2+...+(?n?n, (6) as has already been proved without approximation. 205. To investigate the /re6 motions of the system, we put Qi, Qt, ... Qn = Oyia (5), and assume, in accordance with the usual method of treating linear equations, q^^A^e^^ qt^A^^\ ...q^^A^^* (7) Substituting, we find (aiA« + Ci) A^ + p^Mt + . . . + AnA^n = 0, jSni A^i + j5h, Ail, + . . . + K A» + cj An = 0. Eliminating the ratios Aii A^i ... : Af^,we get the equation a^A* + Ci, Pi^X, . . . ^i«A ftiA, agA* + Cj, ... ^t«A ^nlA, ^n2A, ... anA* + C, = 0, (9) 304 Tidal Waves [chap, vin or, as we shall occasionally write it, for shortness, . A(A) = (10) The determinant A (A) comes under the class called by Cayley 'skew- determinants,* in virtue of the relations (8) of Art. 203. If we reverse the sign of A, the rows and columns are simply interchanged, and the value of the determinant is therefore unaltered. Hence, when expanded, the equation (9) will involve only even powers of A, and the roots will be in pairs of the form A = ± (p -+• i(T). In order that the configuration of relative equilibrium should be stable it is essential that the values of p should all be zero, for otherwise terms of the forms e*^* cos at and e**** sin at would present themselves in the realized expression for any co-ordinate q^. This would indicate the possibility of an oscillation of continually increasing amplitude. • In the theory of absolute equilibrium, sketched in Art. 168, the necessary and sufficient condition of stability (in the above sense) was simply that the potential energy must be a minimum in the configuration of equilibrium. In the present case the conditions are more complicated*, but it is easily seen that if the expression for V — T^ be essentially positive, in other words if the coefficients c^, Cj, . . . c^ in (4) be all positive, the equilibrium must be stable. This follows at once from the equation ® + (F - To) - const., (11) proved in Art. 203, which shews that under the present supposition neither © nor V — Tq can increase beyond a certain limit depending on the initial circumstances. It will be observed that this argument does not involve the use of approximate equations f. Hence stability is assured if F — Tq is a minimum in the configuration of relative equilibrium. But this condition is not essential, and there may even be stability (from the present point of view) with F — T© * maximum, as will be shewn presently in the particular case of two degrees of freedom. It is to be remarked, however, that if the system be subject to dissipative forces, however slight, afEecting the relative co-ordinates g^, q^^ ... g„, the equi- librium will be permanently or * secularly ' stable only if F — Tq is a minimum. It is the characteristic of such forces that the work done by them on the system is always negative. Hence by (6) the expression 2tH- ( F. — Tq) will, so long as there is any relative motion of the system, continually diminish, in the algebraical sense. Hence if the system be started from relative rest in a * They have been investigated by Routh, On ike Stability of a Given State of Motion; Bee also his Advanced Rigid Dynamics, c. vi. t The argument was originally applied to the theory of oscillations about a configuration of absolute equilibrium (Art. 168) by Dirichlet, "Ueber die Stabilit&t des Gleichgewichts," CrelUf t. xxzii. (1846) [Werke, Berlin, 188^97, t ii p. 3]. 205] Ordinary and Seeidar Stability 305 configuration such that V — T^^'iB negative, the above expression, and therefore A fortiori the part F — To* ^^ assume continually increasing negative values, which can only take place by the system deviating more and more from its equilibrium-configuration. This important distinction between * ordinary' or kinetic, and ^seculat* or practical stability was first pointed out by Thomson and Tait*. It is to be observed that the above investigation presupposes a constant angular velocity (a>) maintained, if necessary, by a proper application of force to the rotating solid. When the solid is free, the condition of secular stability takes a somewhat different form, to be referred to later (Chapter xii.). To examine the character of a free oscillation, in the case of stability, we remark that if A be any root of (10), the equations (8) give A,r(A) A«(A)-A«(A) A„(A) "" ^'^^ where A^i, A^, A^j, . . . Ay„ are the minors of any row in the determinant A, and C is arbitrary. It is to be noticed that these minors will as a rule involve odd as well as even powers of A, and so assume unequal values for the two oppositely signed roots (± A) of any pair. If we put A = ± ia, the general symbolical value of g, corresponding to any such pair of roots may be written q, = C^rs (^) e*^ -f C'A„ (- to) e-^. If we put 2A„ (icr) = F, (cr^) + iof. [p% C = K^, C = fCe-**, we get a solution of our equations in real form, involving two arbitrary constants iC, c; thusf q^ = F^ {a^) . K cos (at + c) - cr/i (a^) . K sin (at + c), q^ = F^ (a^) . K cos (at + e) - af^ (a*) . K sin (at + c), ?8 = ^3 (o^) . K cos (at + e)- af^ (a^) . K sin (at + c), ^ (13) g„ = jF„ (a^) . K cos (at + c) - af^ (a») . K sin (at + c). These formulae express what may be called a * natural mode' of oscillation of the system. The number of such possible modes is of course equal to the num]ber of pairs of roots of (10), i.e. to the number of degrees of freedom of the system. ♦ Natural Philosophy (2nd ed.), Part i. p. 391. See also Poincar^, "Sur T^quilibre d*une masse (luide anim^e d'un mouvement de rotation," Acta Mathematical t. vii. (1885), and op. eit ante p. 141. t We might have obtained the same result by assuming, in (6), where A, is real, and rejecting, in the end, the imaginary parts. L. H. 20 306 Tidal Waves [chap, vm If ^, 1), ^ denote the component displacements of any particle from its equilibrium position, we have J. dx , dx dy dy , dx dz (14) y dz dz Substituting from (13), we obtain a result of the form ^ = P . fC cos (a« + c) + P' . fC sin (at + c), ] 1) =- g . Z cos (cr« + €) + e' . fC sin (a« + €), i (15) i = R.K COB (at + €) -\' R' .K sin (at + c), J where P, P', Q, Q\ 22, 22' are determinate functions of the mean position of the particle, involving also the value of a, and therefore dijSerent for the difEerent normal modes, but independent of the arbitrary constants K, e. These formulae represent an elliptic-harmonic motion of period in/a, the directions and t^IL-1 P' Q'~R" • •••••••*• \a.\9 I being those of two conjugate semi-diameters of the elliptic orbit, of lengths (P2 ^ Q2 + jj2)i . X, and (P'a + g'^ ^ 22'«)* . If, respectively. The positions and forms and relative dimensions of the elliptic orbits, as well as the relative phases of the particles in them, are accordingly in each natural mode determinate, the absolute dimensions and epochs being alone arbitrary *. 206. The symbolical expressions for the forced oscillations due to a periodic disturbing force can easily be written down. If we assume that Qi9 Q^y - " Qn ^^ vary as e*^, where a is prescribed, the equations (5) give, if we omit the time-factors. An(tcr)^ . Ai2(*or)^ ^^ A (ta) ^ A (ta) A21 (ia ) ^ , A^Mn A. ^(^a) ^(^a) ■^ A{ia) ^"' Anl (^y) ^ , K2 (^) n A. A^^nn (^) A (ia) A(ia) ^(^a) Qn^ (17) * The theory of the free modes has been further developed by Rayleigh, "On the Free Vibrations of Systems affected with Small Rotatory Terms," Phil. Mag. (6), t. v. p. 293 (1903) [Papers, t. v. p. 89], for the case where the rotatory coefficients /S^, are relatively small. 205-206] Free and Forced OsciUatiom 307 The most important point of contrast with the theory of the 'normal modes' in the case of no rotation is that the displacement of any one type is no longer affected solely by the disturbing force of that type. As a con- sequence, the motions of the individual particles are, as is easily seen from (14), now in general elliptic-harmonic. Again, there are in general differences of phase, variable with the frequency, between the displacements and the force. As in Art. 168, the displacement becomes very great when A (ia) is very small, i.e. whenever the 'speed' a of the disturbing force approximates to that of one of the natural modes of free oscillation. When the period of the disturbing forces is infinitely long, the displace- ments tend to the * equilibrium- values ' ?i — T"> ?2 ——>••• ?n — —•, lio) as is found by putting or = in (17), or more simply from the fundamental equations (5). This conclusion must be modified, however, when one or more of the coefficients of stability c^, Cj, . . . c„ is zero. If, for example, ^1 = 0, the first row and column of the determinant A (A) are both divisible by A, so that the determinantal equation (10) has a pair of zero roots. In other words we have a possible free motion of infinitely long period. The coefficients of Q^y Q3, ... Qn ^^ ^^^ right-hand side of (17) then become indeterminate for a ~ 0, and the evaluated results do not as a rule coincide with (18). This point is of importance, because in some hydrodynamical applications, as we shall see, steady circulatory motions of the fluid, with a constant deformation of the free surface, are possible when no extraneous forces act; and as a consequence forced tidal oscillations of long period do not necessarily approximate to the values given by the equilibrium theory of the tides. Cf. Arts. 214, 217. In order to elucidate the foregoing statements we may consider more in detail the case of two degrees of freedom. The equations of motion are then of the forms <hSi +^1^1 +^J8 =Qi, (hit +c,g, -/9^i =^2 (10) The equation determining the periods of the free oscillations is OjajX* +(aiC3| +a^Ci +/3*) X^+CjC, =0. (20) For 'ordinary' stability it is sufficient that the roots of this quadratic in X' should be real and negative. Since £4, o^ are essentially positive, it is easily seen that this condition is in any case fulfilled if c^, c, ^^^ both positive, and that it will also be satisfied even when Cj , C2 are both negative, provided iS* be sufficiently great. It will be shewn later, however, that in the latter ca«e the equilibrium is rendered unstable by the introduction of dissipative forces. See Art. 316. To find the forced oscillations when Qi, Q2 vb^ as e**^, we have, omitting the time-factor, {Ci-a^ih) gi +i<r^2 =Qi» 'i<^^i +(©2 "O-'^j) gj=Qt» (21) whence a - ' i^-<^<h)Qi'i<rfiQ2 . *V^Oi + (q -cr»a,) Q. Whence ^1 -(c, _^t^) (c, -<,««,) -<r«i3»' ^»"(c, -<r«ai)(c,-<r»a,)-a*/3« ^^^^ 20—2 308 Tidal Waves [chap, viii Let us now suppose that c, =0, or, in other words, that the displacement q^ does not affect the value of V —T^, We will also suppose that Qt=0, i.e. that the extraneous forces do no work during a displacement of the type q^' The above formulae then give In the case of a disturbance of long period we have o- =0, approximately, and therefore '-^T^«- *«=5;^«» <^) The displacement q^ is therefore has than its equilibrium- value, in the ratio 1 : 1 +/3V^^ > and it is accompanied by a motion of the type q^ although there is no extraneous force of the latter type (cf. Art. 217). We pass, of course, to the case of absolute equilibrium, considered in Art, 168, by putting /9=0*. It should be added that the determination of the * principal co-ordinates* of Art. 204 depends on the original forms of ST and V — Tq, and is therefore afEected by the value of a>*, which enters as a factor of Tq* The system of equations there given is accordingly not altogether suitable for a discussion of the question how the character and the frequencies of the respective principal modes of free vibration vary with a>. One remarkable point which is thus overlooked is that types of circulatory motion, which are of infinitely long period in the case of no rotation, may be converted by the sUghtest degree of rotation into oscillatory modes of periods comparable with that of the rotation. Cf. Arts. 212, 223. To illustrate the matter in its simplest form, we may take the case of two degrees of freedom. If c, vanishes for a> =0, and so contains co' as a factor in the general case, the two roots of equation (20) are approximately, when o>' is smalL The latter root makes X x », ultimately. 207. Proceeding to the hydrodynamical examples, we begin with the case of a plane horizontal sheet of water having in the undisturbed state a motion of uniform rotation about a vertical axisf. The results will apply without serious qualification to the case of a polar or other basin, of not too great dimensions, on a rotating globe. Let the axis of rotation be taken as axis of z. The axes of x and y being now supposed to rotate in their own plane with the prescribed angular velocity co, let us denote by w, v, w the velocities at time t, relative to these axeSy of the particle which then occupies the position (x, y, z). The actual velocities of the same particle, parallel to the instantaneous positions of the axes, will be w — oiy, V -f wx, w, and the accelerations in the same directions will be Di - ^^^ - ^ ^' 5^ + ^^ - ^ y' -Dt' * The preceding theory appeared in the 2nd ed. (1896) of this work. t Sir W. Thomson, *'0n Gravitational Oscillations of Rotating Water," Proc, R, 8. Edin^ t. X. p. 92 (1879) [Papere, t. iv. p. 141], 206-207] Two-Dimemional Problems 309 In the present application^ the relative motion is assumed to be infinitely small, so that we may replace D/Dt by d/dt. Now let Zq be the ordinate of the free surface when there is relative equilibrium under gravity alone, so that CO* 2o = i— (^* + y*) H- const., (1) if as in Art. 26. For simplicity we will suppose that the slope of this surface is everywhere very small ; in other words, if r be the greatest distance of any part of the sheet from the axis of rotation, a)^r/g is assumed to be small. If Zq + ( denote the ordinate of the free surface when disturbed, then on the usual assumption that the vertical acceleration of the water is small compared with g, the pressure at any point (x, y, z) will be given by y-yo==^p(2^o+?-2), (2) The equations of horizontal motion are therefore __2a«,= -s,g^-g-, ^ + 2a,«=-<7^--^, ....(3) where Q denotes the potential of the disturbing forces. * If we write f « , (4) 9 these become 3|-2a«,= -4(C-r). |+2a,«=-i,|(C-0. ..(5) The equation of continuity has the same form as in Art. 193, viz. dt" dx dy ' ^' where h denotes the depth, from the free surface to the bottom, in the undisturbed condition. This depth will not, of course, be uniform unless the bottom follows the curvature of the free surface as given by (1). If we eliminate C^Cfrom the equations (5), by cross-differentiation, we find di[^-dj,)*^[^^^)=^' <'' or, writing u^d(/dt, v=dri/dt, and integrating with respect to t, £-5*^(3*1)-"°^ "' This 18 merely the expression of Helmholts' theorem that the product of the vortioity 2» + ^ - R- and the cross-section 1 1 +^ + 5^) ^^y» of a vortex -filament, ib constant. 310 Tidal Waves [chap, vm In the case of a simple-harmonic disturbance, the time-factor being 6^*, the equations (5) and (6) become icm-2cot;=-(7^U~f), iat; + 2a>ti«-(7|^(C-?), ..(9) J . y 3 (Aw) 9 (At?) ,-^. *^^ ^^^--k---V (^^^ From (9) we find (11) and if we substitute from these in (10), we obtain an equation in ( only. In the case of uniform depth the result takes the form Vi*$ + '^^*^ = V,«f, (12) where Vj* = d^jdx^ + a«/9y*, as before. When ^=0, the equationfi (6) and (6) can be satisfied by coMkmi values of u, v, C provided certain oonditions are fulfilled. We must have '*"""2i^' *'-2iS ^^^^ andtherefore ^f^' ^ =0 (14) 9 («, y) The latter condition shews that the contour-lines of the free surface must be everywhere parallel to the contour-lines of the bottom, but that the value of ( is otherwise arbitrary. The flow of the fluid is everywhere parallel to the contour-lines, and it is therefore further necessary for the possibility of such steady motions that the depth should be uniform along the boundary (supposed to be a vertical wall). When the depth is every- where the same, the condition (14) is satisfied identically, and the only limitation on the vcdue of ( is that it should be constant along the boundary. 208. A simple application of the preceding equations is to the case of free waves in an infinitely long uniform straight canal*. Ifweassnme j «= ^6** '«*-*> +"»^ v = 0, (1) the axis of x being parallel to the length of the canal, the equations (9) of the preceding Art., with the terms in f omitted, give cu^gi, 2a}U^-gmC, (2) whilst, from the equation of continuity (Art. 207 (6)), c^^hu (3) We thence derive c^=: gh, m = — 2ct>/c (4) * Sir W. ThomBon, U, ante p. SOS 207-209] Circular Basin 311 The former of these results shews that the wave- velocity is unaffected by the rotation. When expressed in real form, the value of t, is I = ae-^yl^ cos {* (c« - a;) + c} (5) The exponential factor indicates that the wave-height increases as we pass from one side of the canal to the other, being least on the side which is foTwardy in respect of the rotation. If we take account of the directions of motion of a water-particle, at a crest and at a trough, respectively, this result is easily seen to be in accordance with the tendency pointed out in Art. 202*. The problem of determining the free oscillations in a rotating canal of finite length, or in a rotating rectangular sheet of water, has not yet been solved f. 209. We take next the case of a circular sheet of water rotating about its centre t. If we introduce polar co-ordinates r, fl, and employ the symbols fi, to denote displacements along and perpendicular to the radius vector, then since R = iafi, = ia%y the equations (9) of Art. 207 are equivalent to whilst the equation of continuity (10) becomes d(hRr) d(h%) *"" rdr rde ^^' Hence ^^^ - w (a; " Vfae) ^^ ~ ^^' ® ° a> -"w Wd^ ~ *i^) ^^ ~ ^' (3) and substituting in (2) we get the differential equation in £. In the case of uniform depth we find ( V + K«) J = V,»^, (4) -<i ''*=^^' (') This might have been written down at once from Art. 207 (12). * For applications to tidal phenomena see Sir W. Thomson, Nature, t. xiz. pp. 154, 671 (1879). t Except in the case where the angular velocity of rotation is relatively small. For this see Rayleigh, "On the Vibrations of a Rectangular Sheet of Rotating Liquid," Phil Mag. (6), t. v. p. 297 (1903) [Papers, t. v. p. 93]. X The investigation which follows is a development of some indications given by Kelvin in the paper cited on p. 308. 312 Tidal Waves [chap, vra The condition to be satisfied at the boundary (r = o, say) is 22 = 0, or a 2to a ('!—»)«-£'-» <" 210.- In the case of the free oscillations we have f = 0. The way in which the imaginary i enters into the above equations, taken in conjunction with Fourier's Theorem, suggests that occurs in the form of a factor 6***, where 8 is integral. On this supposition, the differential equation (4) becomes dr* ^Yhi^-T^i-" <«) and the boundary-condition (7) gives 4^^C = 0. ..(9) for r ^ a. The equation (8) is of Bessel's form, and the solution which is finite for r = may therefore be written i = AJ, (kt) e'^^*+^ ; (10) but it .is to be noticed that k^ is not, in the present problem, necessarily positive. When k* is negative, we may replace J. {kt) by Z, (/cV), where K is the positive square root of (4co* — <^)l9K and • ^^^ " 2* . » ! 1 ^ 2 (2« + 2) ^ 2 . 4 (2« + 2) (2s + 4) "^ • • • J * "^^ In the case of symmetry about the axis (s = 0), we have, in real form, ^ = ^JoM-cos(a« + €), (12) where k is determined by Jo'('ca) = (13) The corresponding values of a are then given by (6). The free surface has, in the various modes, the same forms as in Art. 191, but the frequencies are now greater, viz. we have (7« = V + 4a>«, (14) where Oq is the corresponding value of a when there is no rotation. It is easily seen, moreover, on reference to (3), that the relative motions of the fluid particles are no longer purely radial ; the particles describe, in fact, ellipses whose major axes are in the direction of the radius vector. For « > we have i = AJ^ (kt) . cos (or« + «& + €), (15) where the admissible values of /c, and thence of a, are determined by (9), which gives koJ: (ko) + — J, {Ka) = (16) * The fonotions /, (2) were tabulated by Prof. A Lodge, Brit Asa, Rep. 1889. The tables are reprinted by Dale, and by Jahnke and Emde. 209-210] Circular Basin 313 The fonniila (15) represents a wave rotating relativdy to the water with an angular velocity a/«, the rotation of the wave being in the same direction with that of the water, or the opposite, according as a/a> is negative or positive. Some indioatioDB as to the values of <r may be gathered from a graphical oonstruction. If we write jc'a' =a?, we have, from (6), ^=4^1)*' ^"^ where If we farther put the equation (16) may be written 3 = icaJ/ (ko) gh =<t>{ic^a*). .(18) <l>{x)± (-1)*- .(19) .(20) The curve y = - ^ (a?) can be readily traced by means of the tables of the functions Jg (z), /« (z) ; and its inter- sections with the parabola y« = l+ar//3 ,.(21) will give, by their ordinates, the values of cr/2oi>. The constant jS, on which the positions of the roots depend, is equal to the square of the ratio 2<oal{gh)^ which the period of a wave travelling roimd a circular canaJ of depth h and perimeter 2ira bears to the half -period {ir/») of the rotation of the water. The accompanying figures indicate the relative magnitudes of the lower roots, in the oases « = 1 and 8=2, when /9 has the values 2, 6, 40, respectively*. p.2 * For clearness the scale of y has been taken to be 10 times that of z. 314 Tidal Waves [CHAP, vin With the help jof these figures we can trace, in a general way, the changes in the character of the fiee modes as /9 increases from zero. The results may be interpreted aa due either to a continuous increase of «, or to a continuous diminution of A. We will use [.=2] the terms 'positive' and 'negative' to distinguish waves which travel, relatively to the water, in the same direction as the rotation and the opposite. When /3 is infinitely small, the values of x are given by J, (x^) =0; these correspond to the vertical asymptotes of the curve (20). The values of o- then occur in pairs of equal and oppositely-signed quantities, indicating that there is now no difference between the velocity of positive and negative waves. The case is, in fact, that of Art. 191 (12). As jS increcbses, the two values of <r forming a pair become unequal in magnitude, and the corresponding values of x separate, that being the greater for which (r/2ci> is positive. When /3 =«(« + !) the curve (20) and the parabola (21) Umch at the point (0, — 1)» the corresponding value of o- being —2», As fi increases beyond this critical value^ one value of x becomes negative, and the corresponding (negative) value of cr/2» becomea smaller and smaller. Hence, as jS increases from zero, the relative angular velocity becomes greater for a negative than for a positive wave of (approximately) the same type; moreover the value of (T for a negative wave is always greater than 2«. As the rotation increases, the two kinds of wave become more and more distinct in character as well as in 'speed.' With a sufficiently great value of /3 we may have one, but never more than one, positive wave for which cr is numerically less than 2a>. Finally, when /3 is very great, the value of o- corresponding to this wave becomes very small compared with «, whilst the remaining values tend all to become more and more nearly equal to ±2». 210-211] Circular Basin 315 If we use a zero suffix to distinguish the ease of <o =0, we find <r«_ ic'+4a)VyA _a;+/3 _ "";:"» y^^f VQ <0 *0 where x^ refers to the proper asymptote of the curve (20). This gives the 'speed' of any free mode in terms of that of the corresponding mode when there is no rotation. 211. As a sufficient example oi forced oscillations we may assume f = Q' e<<'*+'«+'», (23) where the value of a is now prescribed. This makes V^'f "^ 0> ^^^ ^^^ equation (4) then gives J = AJ, {kt) e<«^«+'«+'), (24)' where il is to be determined by the boundary-condition (7), viz. 2a> 8 ('-t) A i ^ .0 (25) o • This becomes very great when the frequency of the disturbance is nearly coincident with that of a free mode of corresponding type*. From the point of view of tidal theory the most interesting cases ore those of « = 1 with cr=oi>, and ^=3 2 with <r=2<0, respectively. These would represent the diurnal and semidiurnal tides due to a distant disturbing body whose proper motion may be nciglected in comparison with the rotation ». In the case of « = ! we have a ttm/ofm horizontal disturbing force. Putting, in addition, o- = <», we find without difficulty that the amplitude of the tide-elevation at the edge (r =a) of the basin has to its * equilibrium- value' the ratio A(z)+2/o(2) ^^^^ where z^\ n/(3/3). With the help of Lodge's tables we find that this ratio has the values 1-000, -638, -396, for j3=: 0, 12, 48, respectively? When (T =2o>, we have k =0, and thence, by (23), (24), (25), f=f, (27) i.e. the tidal elevation has exactly the equilibrium- value. This remarkable result can be obtained in a more general manner; it holds whenever the disturbing force is of the type f=;^(r)e««»*+^+*) (28) provided the depth A be a function of r only. If we revert to the equations (1)» we notice * The case of a neaWy circular sheet is treated by Proudman, " On some Cases of Tidal Motion on Rotating Sheets of Water," Froc. Lond. Maih. 8oc. t. xii. p. 463 (1913). 316 Tidal Waves [chap, vin that when cr=2tt they are satisfied by C=C* O^^-S* To determine i2 as a function of r, we substitute in the equation of continuity (2), which gives • •«•■ ■ * ••■• • ^^-t^l^^-^^r) (29) The arbitrary constant which appears on integration of this equation is to be determined by the boundary-condition. In the present case we have x(^)=^^/a'* Integrating, and making 12=0 -for r=a, we find ^ij=^'(a«-r«) «'<•-*+••+•> (30) The relation Q=%R shews that the amplitudes of R and e are equal, while their phases differ by 00° ; the relative orbits of the fluid particles are in fact circles of radii '=2*5^(»'-'*) <»1) described each about its centre with angular velocity 2» in the negative direction. We may easily deduce that the path of any particle in space is an ellipse of semi-axes r±T described about the origin with harmonic motion in the positive direction, the period being 2irla.^ This accounts for the peculiar features of the case. For if ( have always the equilibrium- value, the horizontal forces due to the elevation exactly balance the disturbing force, and there remain only the forces due to the undisturbed form of the free surface (Art. 207 (1)). These give an acceleration gdzjdr, or <h>V, to the centre, where r is the jradius vector of the particle in its actual position. Hence all the conditions of the problem are satisfied by elliptic-harmonic motion of the individual particles, provided the positions, the dimensions, and the 'epochs' of the orbits can be adjusted so as to satisfy the condition of continuity, with the assumed value of (» The investigation just given resolves this point. 212*. We may also briefly notice the case of a circular basin of variable depth, the law of depth being the same as in Art. 193, viz. * = *o(l-J.) (1) Assuming that i^, 6, ^ all vary as e'*'*+**+*^ and that A is a function of r only, we find, from Art 209 (2), (3), Introducing the value of h from (1), we have, for theJVee oscillations, Tins is identical with Art. 193 (6), except that we now have ghQ aa* in place of cr^/gh^. The solution can therefore be written down from the results of that Art., viz. if we put __ _=n(n-2)-^ (4) * See the footnote to Art. 193. 211-212] Badn of Variable Depth 317 wehave C =4. Q' J (a, A y. ^,) e* "*+--^-' (6) where 'a='|nH^J4, j8 = l+J«— J«, ystf + l; and the condition of oonveigence at the boundary r =a requires that »=»+2j (6) where j is some poflitive integer. The values of o- are then given by (4). The forms of the free surface are therefore the same as in the ease of no rotation, but the motion of the water-particles is different. The relative orbits are in fact now ellipses having their principal axes along and perpendicular to the radius vector; this follows easily from Art. 209 (3). In the symmetrieal modes (a =0), the equation (4) gives cr« = <ro« + 4««, (7) where o-q denotes the 'speed' of the corresponding mode in the case of no rotation, as found in Art. 193. For any value of a other than zero, the most important modes are those for which n=« + 2. The equation (4) is then divisible by cr+2<io, but this is an extraneous factor; discarding it, we have the quadratic «r«-2fi>(r=2«?^°, (8) (9) whence o- =» ± f «* +2« -^ j This gives two waves rotating round the origin, the relative wave-velocity being greater for the negative than for the positive wave, as in the case of uniform depth (Ait. 210). With the help of (8) the formulae reduce to f=^.(9'. ^=^i^.($j'\ «=*»£^.©'" <»«) the factor e' '''*'*"**'*^^ being of course understood in each case. Since e=ti?, the relative orbits are all circles. The case « = 1 is noteworthy; the free surface is then always plane, and the circular orbits have all the same radius. When n>« +2, we have nodal circles. The equation (4) is then a cubic in 0-/20); it is easily seen that its roots are all real, lying between — oo and —1, —1 and 0, and +1 and + 00 , respectively. As a numerical example, in the case of « = 1, n =6, corresponding to the values 2, 6, 40 of ^*a^lg\, we find ( + 2-889 + 1-874 + M80, - 0-125 -0-100 -0-037, -2-764 -1-774 -1-143. The first and the last root of each triad give positive and negative waves of a somewhat similar character to those already obtained in the case of uniform depth. The smaller negative root gives a comparatively slow oscillation which, when the angular velocity o> is infinitely small, becomes a steady rotational motion, without elevation or depression of the surface*. * The possibility of oscillations of this type was pointed out in Art. 206, ad fin. 318 Tidal Waves [chap, vni The most important type oi forced oscillations is such that f=0gYe<<'*+'*-^> (11) We readily verify, on substitution in (3), that '"'2«^Ao~(cr*-2«ir)a«^ ^^^^ We notice that when crs2«> the tide-height has exactly the equilibrium-value, in agree- ment with Art. 211. If (Ti, cTi denote the two roots of (8), the last formula may be written ^"(l-cr/crjd-cr/cr,) <^^) Tides on a Rotating Globe. 213. We proceed to give some account of Laplace's problem of the tidal oscillations of an ocean of (comparatively) small depth covering a rotating globe*. In order to bring out more clearly the nature of the approximations which are made on various groimds, we adopt a method of establishing the fundamental equations somewhat different from that usually followed. When in relative equilibrium, the free surface is of course a level-surface with respect to gravity and centrifugal force; we shall assume it to be a surface of revolution afcout the polar axis, but the ellipticity will not in the first instance be taken to be small. We adopt this equilibrium-form of the free surface as a surface of reference, and denote by 6 and <f> the co-latitude (i.e. the angle which the normal makes with the polar axis) and the longitude, respectively, of any point upon it. We shall further denote by z the altitude, measured outwards along a normal, of any point above this surface. The relative position of any particle of the fluid being specified by the three orthogonal co-ordinates 0, <^, z, the kinetic energy of unit mass is given by 2T^(R-{-zYd^ + tn^{oy^4>Y-\-z\ (1) where 2?' is the radius of curvature of the meridian-section of the surface of reference, and m is the distance of the particle from the polar axis. It is to be noticed that i2 is a function of d only, whilst to is a function of both 6 and z ; and it easily follows from geometrical considerations that = cos ff, -5- = sm (2) (R-\-z)dd ' dz * "Recherches sur quelquee points du syst^me du monde," Mim, de VAcad, toy, de» Sciences, 1775 [1778] and 1776 [1779]; Oeuvres Completes, t. ix. pp. 88, 187. The mvestigation is repro- duced, with various modifications, in the Micanique Celeste, livre 4"**, c. i. (1799). 212-213] Tides on a Rotating Globe 319 The component accelerations are obtained at once from (1) bj Lagrange's formula. Omitting terms of the second order, on account of the restriction to infinitelj small motions, we have 1 fddT dT\ ,D^ .x 1 / , lo ix ^o* 1 fd dT dT\ 1 ^ ^ f^m A , dm A \ ..(3) dt dz dz dz' Hence, if we write u, v, w for the component relative velocities of a particle, viz. U'={R + z)0, v = w^, w^z, (4) and make use of (2), the hydrodynamical equations may be put in the forms |^-2a«;cose . . ^^| (? + ^ . J^.^a + q)/ 5^4- 2ci>ttcos&+ 2a>wsine= - ~ ^ f- + ^ - W^^ + Q), I., (5) ^-2a>t;sinfl = ~ ^ (? + ^ - Jco«m« + q), where ^ is the gravitation-potential due to the earth's attraction, whilst £2 denotes the potential of the disturbing forces! So far the only approximation has consisted in the omission of terms of the second order in ti, t;, w. In the present application, the depth of the sea being small compared with the dimensions of the globe, we may replace R-\- z by fi. We will further assume that the vertical velocity w is small compared with the horizontal components u, v, and that dwjdt may be neglected in comparison with oyv. As in the theory of 'long' waves, such assumptions are justified a posteriori if the results obtained are found to be consistent with them (cf. Art. 172). Let us integrate the third of equations (5) between the limits z and £, where J denotes the elevation of the disturbed surface above the surface of reference. At the surface of reference (z «= 0) we have ^ — Jco^tu* = const., by hypothesis, and therefore at the free surface {z^ t) \ff ^ ^w^xn^ = const. + ^{, (6) approximately, provided ^ = g- (^ — Jcu^id*) Here g denotes the value of apparent gravity at the surface of reference; it is of course, in general, a function of 0, but its variation with z is neglected. 320 Tidal Waves [Chap, vin *The integration in question then gives 2 + >I^ - \a>^w^ = const. + ^f + 2a) sin & | vdz, (7) P J e where the variation of the disturbing potential Q with z has been neglected in comparison with g. The last term is of the order of whv sin d, where h is the depth of the fluid, and it may be shewn that in the subsequent applications this is of the order A/a as compared with g^. Hence, substituting in the first two of equations (5), we obtain, with the approximations indicated, ^-2««;co80=-A<,(^-r). | + 2«,«cose=-^^(S-f),..(8) where f=-£i/^ (9) These equations are independent of z, so that the horizontal motiou may be assumed to be sensibly the same for all particles in the same vertical line. As in Art. 198, this last result greatly simplifies the equation of continuity. In the present case we find without difficulty d^_ i( d jhwu) d (hv)] dt~ m\ Rde "^ d<l> ] ^ ^ It is important to notice that the preceding equations involve no assumptions beyond those expressly laid down; in particular, there is no restriction as to the ellipticity of the meridian, which may be of any degree of oblateness. 214. In order, however, to simplify the question as far as possible, without sacrificing any of its essential features, we now take advantage of the circumstance that in the actual case of the earth the ellipticity is a small quantity, being in fact comparable with the ratio {lo^ajg) of centrifugal force to gravity at the equator, which ratio is known to be about ^. Subject to an error of this order of magm'tude, we may put fi = a, id = a sin fl, g = const., where a is the earth's mean radius. We thus obtain |-2a..cos^=-f|(C--r), |+2a>«COB^=-f^^(J-a (1) with dt^ 1 ■ dt a sin d d (hu sin d) d (hv) + ^- , (2) dd ' d<i> this last equation being identical with Art, 198 (1)*. Some conclusions of interest foUow at once from the mere form of the equations (1). In the first place, if u, v denote the velocities along and * Except for the notation these are the equations arrived at by Laplace, tc. ante p. 318. 213-215] Tides on a Rotating Globe 321 perpendicular to any horizontal direction 8, we easily find, by transforma- tion of co-ordinates |^-2a,vco8d=-<7|(C-?) (3) In the case of a narrow canal, the transverse velocity v is zero, and the equation (3) takes the same form as in the case of no rotation ; this has been assumed by anticipation in Art. 183. The only effect of the rotation in such cases is to produce a slight slope of the wave-crests and furrows in the direction across the canal, as investigated in Art. 208. In the general case, resolving at right angles to the direction of the relative velocity {q, say), we see that a fluid particle has an apparent acceleration 2o)q cos d towards the right of its path, in addition to that due to the forces. Again, by comparison of (1) with Art. 207 (5), we see that the oscillations of a sheet of water of relatively small dimensions, in colatitude d, will take place according to the same laws as those of a plane sheet rotating about a normal to its plane with angular velocity ay cos d. As in Art. 207, free steady motions are possible, subject to certain conditions. Putting ^=0, we find that the equations (1) and (2) are satisfied by constant values of w, v, J, provided u_ ^ ^^ v^ ^ ^^ (4) 2a>a sin d cos 6 d<f) ' 2a>a cos Odd^ '-^^mw-" «>' The latter condition is satisfied by any assumption .of the form S=/(Asecfl) (6) and the equations (4) then give the values of u, v. It appears from (4) that the velocity in these steady motions is everywhere parallel to the contour- lines of the disturbed surface. If h is constant, or a function of the latitude only, the only condition imposed on t, is that it should be independent of <f)\ in other words the elevation must be symmetrical about the polar axis. 215. We shall suppose henceforward that the depth A is a function of B only, and that the barriers to the sea, if any, coincide with parallels of latitude. We take first the cases where the disturbed form of the water-surface is one of revolution about the polar axis. When the terms involving <f> are omitted, the equations (1) and (2) of the preceding Art. take the forms ^ - 2o)V cos = - - ^ (^ - f), X- + %JDU cos & = 0, (1) 01 a ou ut with i=-— -t.? (2) at a sm Odd L. H. 21 322 Tidal Waves [chap, vin Aflsuming a time-factor 6^^ and solving for u, v, we find a« - 4a>« cos« 0add^^ ^^' ^ " (t« - 4coa cos* dadd^^" ^^' " ' ^^ .., , y d (Aw sin 0) ... with taj = - — .-- o^ (4) The formulae for the component displacements (f, iy, say) can be written down from the relations w = f , t; = ^, or w = taj, v = icny. It appears that the fluid particles describe ellipses having their principal axes along the meridians and the parallels of latitude, respectively, the ratio of the axes being a/2a} . sec 6, In the forced oscillations of the present type the ratio a/2o} is very small ; so that the ellipses are very elongated, with the greatest length from E. to W., except in the neighbourhood of the equator. Eliminating u and t; between (3) and (4), and writing, for shortness, In the case of uniform depth, this becomes |;(^.|)h-«'-« (') where /a = cos 0, and j3 = -r— = — , - (8) 216. First, as regards the/re6 oscillations. Putting f = 0, we have I; (t^. a^) + « - ». <') and we notice that in the case of no rotation this is included in (1) of Art. 199, as may be seen by putting Pf^ = (J^a^/gh,f = oo . The general solution of (9) is necessarily of the form C = ^l* {/*) + 5/ (m), (10) where F {fi) is an even, and / (fi) an odd, function of ^, and the constants A, B are arbitrary. In the case of a zonal sea bounded by two parallels of latitude, the ratio A : B, and the admissible values of / (and thence of the frequency a/2n) are determined by the conditions that m = at each of these parallels. If the boundaries are symmetrically situated on opposite sides of the equator, the oscillations fall into two classes; viz. in one of these B = 0, and in the other ^ = 0. By supposing the boundaries to contract to points at the poles, we pass to the case of an unlimited ocean, and the admissible values of / are now determined by the condition that u must be finite f or ^ = ± 1. The argument is, in principle, exactly that of Art. 201, 215-216] Case of Symmetry 323 but the application of the last-mentioned condition is now more difficult, owing to the less familiar form in which the solution of the differential equation is obtained. In the case of symmetry with respect to the equator, we assume, following the method of Kelvin* and Darwin t, ^^^4r^|^ = Bi^ + B3M»+...+Ba/+i/^«+^ (11) This leads to (12) where A is arbitrary ; and makes (13) Substituting in (9), and equating coefficients of the several powers of /t, we find B^-^A = (14) ^3 - (l - If^) 5i = 0, (15) and thenceforward ^2^+1 - (l - 2i (2] + 1)) ^*^-^ " ^W+^ ^^'^ " ^ ^^^^ These equations determine B^^B^, ... B^s^^y ... in succession, in terms of A, and the solution thus obtained would be appropriate, as already explained, to the case of a zonal sea bounded by two parallels in equal N. and S. latitudes. In the case of an ocean covering the globe, it would, as we shall prove, give infinite velocities at the poles, except for certain definite values of/. Let us write ^ay+i/^a^-i = -^i+iJ (17) we shall shew, in the first place, that as j increases N^ must tend either to the limit or to the limit 1. The equation (16) may be written "^'^^ 23{23-hiy2j(23 + \)N, ^^^^ Hence, when j is large, either ^*^~ 2j"(2; + l)' (^^) * Sir W. Thomson, "Note on the 'Oscillations of the First Species' in Laplace's Theory of the Tides," PhU, Mag. (4), t. L p. 279 (1876) [Papers, t. iv. p. 248]. t "On the Dynamical Theory of the Tides of Long Period," Proe. Roy, 8oc, t. xlL p. 337 (1886) [Papers, t. i. p. 366], 21—2 324 Tidal Waves [ohap. vm approximately, or N^^^ is not small, in which case N^^^ ^^ ^^ nearly equal to 1, and the values of iV^+8> ^i+4> • • • ^^ ^^^^ more and more nearly to 1, the approximate formula being ^^+^=^~2i(2i+l) ^^^^ Hence, with increasing y, N^ tends to one or other of the forms (19) and (20). In the former case (19), the series (11) will be convergent for ^ = ± 1, and the solution will be valid over the whole globe. In the other event (20), the product N^N^ ... JVy+i, and therefore the. coefficient 52/+i> tends with increasing ^ to a finite limit other than zero. The series (11) will then, after some finite number of terms, become com- parable with 1 4- /A* + ^* 4- . . . , or (1 — /x*)~^, so that we mav write ■ ' «'-i+.^ ' m where L and M are functions of /a which remain finite when /a »= ± 1. Hence, from (3), »=-£7^* I'— £'"-'->*^+(>-''*)"' ■«>•■•" which makes u infinite at the poles. It follows that the conditions of our problem can be satisfied only if Nf tends to the limit zero ; and this consideration, as we shall see, restricts us to a determinate series of values of/. The relation (18) may be put in the form ff, "W^ (23) 1 PJ _ AT and by successive applications of this we obtain N^ in the form of a convergent continued fraction j3 _ _p ^ ~2i(2i+l) (2j+ 2yW+"3) (2.7 + 4) {2j + 5) PP ] W \ j3/' ^~2/(2j>I)"^ ^■"(2j+2)(2j + 3)+ ^~(2i+4)(2; + 5)"^-" (24) on the present supposition that Nj^j^ tends with increasing k to the limit 0, in the manner indicated by (19). In particular, this formula (24) determines the value of JVj. Now from (15) we must have iV,= l-|4' (25) 216-217] Free OsdUations 326 which is equivalent to iV^ = oo . This equation determines the admissible values of /{= (T/2a)). The constants in (11) are then given by Bi = pA, Ba = N^PA, B^ = N^N^PA, . . . , .... (27) where A is arbitrary. It is easily seen that when j3 is infinitesimal the roots of (26) are given by a^a^ = i3/« = w(n+l), (28) gh where n is an even integer ; cf . Art. 199. One arithmetically remarkable point remains to be noticed. It might appear at first sight that when a value of / has been found from (26) the coefficients B^, B^, B^y ... could be found in succession from (15) and (16), or by means of the equivalent formula (18). But this would require us to start with exactly the right value of / and to observe absolute accuracy in the subsequent stages of the work. The above argument shews, in fact, that any other value, difiering by however little, if adopted as a starting point for the calculation will inevitably lead at length to values of Nf which approximate to the limit 1*. 217. It is shewn in the Appendix to this Chapter that the tide- generating potential, when expanded in simple-harmonic functions of the time, consists of terms of three distinct types. The first type is such that the equilibrium tide-height would be given by f = JBT ' (i - COS* 0) . cos ((T« -F €) t (29) The corresponding forced waves are called by Laplace the ' Oscillations of the First Species' ; they include the lunar fortnightly and the solar semi-annual tides, and generally all the tides of long period. Their characteristic is symmetry about the polar axis, and they form accordingly the most important case of forced oscillations of the present type. If we substitute from (29) in (7), and assume for 1^* ^' and U * Sir W. Thomson, le, ante p. 323. t In strictnefis, 6 heie denotes the geocentric latitude, but the difference between this and the geographical latitude may be neglected consistently with the assumptions introduced in Art. 214. 326 Tidal Waves [chap, vm expressions of the forms (11) and (12), we have, in place of (14), (15), 5, - i/S^' - /3^ = (30) /3/« 5.-(l-|^)5i + ii8£r' = 0, (31) whilst (16) and its consequences hold for all the higher coefficients. It may be noticed that (31) may be included under the general formula (16), provided we write B_^ = — 2H\ It appears by the same argument as before that the only admissible solution for an ocean covering the globe is the one that makes N^ = 0, and that accordingly iV^^ must have the value given by the continued fraction in (24), where /is now prescribed by the frequency of the disturbing forces. In particular, this formula determines the value of N^. Now and the equation (30) then gives A^-^H'-^N^W; (32) in other words, this is the only value of A which. is consistent with a zero limit of Nj, and therefore with a finite velocity at the poles. Any other value of A, if adopted as a starting point for the calculation of jB^, £3, £5, ... in succession, by means of (30), (31), and (16), would lead ultimately to values of Ni approximating to the limit 1. Moreover, since abaoVute accuracy in the initial choice of A and in the subsequent computations would be essential to avoid this, the only practical method of calculating the coefficients is to use the formulae B^H' = - 2iVi, £3 = N^Bj^, B, = N^B^, . . . , or BJH' = -2N^, B^jH' = -2N^N^, BJH' ^ ^2N^N^N^, .. . (33) where the values oi Ni, N^, N^y ... are to be computed from the continued fraction (24). It is evident a posteriori that the solution thus obtained will satisfy aH the conditions of the problem, and that the series (12) will converge with great rapidity. The most convenient plan of conducting the ctdculation is to assume a roughly approximate value, suggested by (19), for one of the ratios N^ of sufficiently high order, and thence to compute in succession by means of the formula (23). The values of the constants AjBiyB^, . . . , in (12), are then given by (32) and (33). For the tidal elevation we find ^/H' = - 2NJ^ - (1 -f*N,) ,*» - iN, (1 -/W.) /*«-... - -. NiNt . . . Ni_i (1 -f*Ni) /*« - (34) 217] Tides of Long Period 327 In the case of the lunar fortnightly tide, / is the ratio of a sidereal day to a lunar month, and is therefore equal to about ^, or more precisely '0365. This makes/* = '00133. It is evident that a fairly accurate representation of this tide, and d, fortiori of the solar semi-annu^ tide, and of the remaining tides of long period, will be obtained by putting / = ; this materially shortens the calculations. The results will involve the value of j8, = 4ai*a*/^A. For j8 = 40, which corresponds to a depth of 7260 feet, we find in this way ijE' = -1515 - l'0000/i« + 1'5153/A* - l'2120/i« + '6063/^8 - •2076/ii<> + -0516/^" - 0097/1" + OOlS/Lii* - 0002/1", (35)* whence, at the poles (/a = ± 1), i = - f ff ' X '154, and, at the equator (/i = 0), S = iff ' X -455. Again, for jS = 10, or a depth of 29040 feet, we get IjH' = '2359 - lOOOO/i* + -5898/1* - '1623/i« + -0258/18 - 0026/110 -f -0002/112 (36) This makes, at the poles, ^=-|ff' X '470, and, at the equator, • {= iff ' X '708. For jS = 5, or a depth of 58080 feet, we find IjH' = -2723 - lOOOO/i* + '3404/i* - '0509/i« + '0043/i8 - -0004/iio (37) This gives, at the poles, S = - Iff ' X -651, and, at the equator, i = iff ' X '817. Since the polar and equatorial values of the equilibrium tide are — fff ' and iff', respectively, these results shew that for the depths in question the long-period tides are, on the whole, direct, though the nodal circles will, of course, be shifted more or less from the positions assigned by the equi- librium theory. It appears, moreover, that, for depths comparable with the actual depth of the sea, the tide has less than half the equilibrium value. It is easily seen from the form of equation (7) that with increasing depth, and consequent diminution of j3, the tide-height will approximate more and more closely to the equilibrium value. This tendency is illustrated by the above numerical results. * The ooeffioiente in (35) and (36) differ only slightly from the numerical values obtained by Darwin for the case /= '0365. 328 Tidal Waves [CHAP, vm It is to be remarked that the kinetic theory of the long-period tides was passed over by Laplace, under the impression that practically, owing to the operation of dissipative forces, they would have the vtdues given by the equilibrium theory. He proved, indeed, that the tendency of frictional forces must be in this direction, but it has been pointed out by Darwin* that in the case of the fortnightly tide, at all events, it is doubtful whether the effect would be nearly so great as Laplace supposed. We shall return to this point later. 218. When the disturbance is no longer restricted to be symmetrical about the polar axis, we must recur to the general equations (1) and (2) of Art. 214. We retain, however, the assumptions as to the law of depth and the nature of the boundaries introduced in Art. 215. If we assume that i2, w, v, { all vary as e<f<'*+^+«), where $ is integral, the equations referred to give icFU — %A)V COS fl = — - ;^ ({ — ?), icFV -\- 2cott cos fl = — . - ....(1) with Solving for w, v, we find . ^ _ 1 (3 (hu sin B) , . -fwAvl (2) 4m (/2 — cos^ a /cosfl3J' . w f\ (3) where we have written as before. 2a> =/. o}^a = m, (4) It appears that in all cases of simple-harmonic oscillation the fluid particles describe ellipses having their principal axes along the meridians and parallels of latitude, respectively. Substituting from (3) in (2) we obtain the differential equation in J' : _ ., ^ .fl flcot e^^ + «»r cosec* e) + ifmi' = - ima^. . . (6) /* — cos* d\f 08 I * l.c. anU p. 323. 217-219] Diurnal Tides 329 219. The case « » 1 includes, as forced oscillations, Laplace's ' Oscillations of the Second Species,' where the disturbing potential is a tesseral harmonic of the second order ; viz. f «- ff " sin fl cos fl . cos (a« H- ^ + €), (1) where a differs not very greatly from w. This includes the limar and solar diurnal tides. In the case of a disturbing body whose proper motion could be neglected, we should have a = a>, exactly, and therefore/ = J- In the case of the moon, the orbital motion is so rapid that the actual period of the principal lunar diurnal tide is very appreciably longer than a sidereal day*; but the sup- position that/= ^ simplifies the formulae so materiaUy that we adopt it in the following investigation f. We find that it enables us to ccJculate the forced oscillations when the depth follows the law A = (1 - gr COS* fl) Ao, (2) where q is any given constant. Taking an exponential factor 6*<'**+*+*\ and therefore putting * = 1,/= i, in Art. 218 (3), and assuming r=Csinflcosfl, (3) C C we find w=— t<7— , v = a— . cos (4) mm Substituting in the equation of continuity (Art. 218 (2)), we get £'+?-^S (^' which is consistent with the law of depth (2), provided ^==~l-2?Ao/ma^" ^^^ Thisgives ^^_ 2gVma ^ ^ ^ 1 - 2qhQ/ma * ^ ' One remarkable consequence of this formula is that in the case of uniform depth {q = 0) there is no diurnal tide, so far as the rise and fall of the surface is concerned. This result was first estabb'shed (in a different manner) by Laplace, who attached great importance to it as shewing that his kinetic theory was able to account for the relatively small values of the diurnal tide * It is to be remarked, however, that there is an important term in the harmonic develop- ment of for which <r = ta exactly, provided we neglect the changes in the plane of the disturbing body's orbit. This period is the same for the sun as for the moon, and the two partial tides thus produced combine into what is caUed the *luni-solar* diurnal tide. t Taken with very slight alteration from Airy, "Tides and Waves," Arts. 96 ... , and Darwin, Eneye, BriU 9th ed., t. xxiii. p. 359. 330 Tidal Waves [chap, vm which are given by observation, in striking contrast to what would be demanded by the equilibrium theory. But, although with a uniform depth there is no rise and fall, there are tidal currents. It appears from (4) that every particle describes an ellipse whose major axis is in the direction of the meridian, and of the same length in all latitudes. The ratio of the minor to the major axis is cos0, and so varies from 1 at the poles to at the equator, where the motion is wholly N. and S. 220. In* the case a = 2, the forced oscillations of most importance are where the disturbing potential is a sectorial harmonic of the second order. These constitute Laplace's * Oscillations of the Third Species,' for which f = H'" sin«fl . cos (a« + 2^ + €), (1) where a is nearly equal to 2a>. This includes the most important of all the tidal oscillations, viz. the lunar and solar semi-diurnal tides. If the orbital motion of the disturbing body were infinitely slow we should have a = 2co, and therefore/ = 1 ; for simplicity we follow Laplace in making this approximation, although it is a somewhat rough one in the case of the principal lunar tide*. A solution similar to that of the preceding Art. can be obtained for the special law of depth f A = Aosin*fl (2) Adopting an exponential factor e*l*"*+**+*^ and putting therefore/ = 1, « = 2,. we find that if we assume r = Csin^fi, (3) the equations (3) of Art. 218 give u = — C cot 6, t?=-s-C,- r-^, (4) m 2m 1 + cos^fi ^ ' whence, substituting in Art. 218 (2), i = ?^.Csin«fi (5) ma ^ ' Putting C = r + t> ^^^ substituting from (1) and (3), we find C^-', ^-r-H"\ (6) 1 — 2hQ/ma ^ ' and therefore C=- . ^V/T I (7) * There is, however, a *lniu-8olar' semi-diomal tide whose speed is exactly 2w if we neglect the changes in the planes of the orbits. Cf. p. 329, first footnote. t Cf. Airy and Darwin« U.ee, 219-221] Semi-diurnal Tides 331 For such depths as actually occur in. the ocean 2Ao < ma, and the tide is therefore inverted. It may be noticed that the formulae (4) make the velocity infinite at the poles, as was to be expected, since the depth there is zero. 221. For any other law of depth a solution can only be obtained in the form of a series. In the case of uniform depth we find, putting * = 2,/= 1, ima/h = j3 in Art. 218 (5), ^^ ~ ^'^'l^ + {^(1 - H'')' - 2,.« - 6} r = - i8(l - firl ..(8) where /i is written for cos 0. In this form the equation is somewhat intract- able, since it contains terms of four different dimensions in /i. It simplifies a little, however, if we transform to V, =(1 -,.«)*, =sinfl, as independent variable ; viz. we find ^i^-^) ^5' -v^-{8-2v»- /3v«) ^' = - j8,^f = - ^H'" A . .(9) which is of three different dimensions in v. To obtain a solution for the case of an ocean covering the globe, we assume $' = Bo + B^v* + B^iA + . . . + B^v^ + (10) Substituting in (9), and equating coeflScients, we find Bo = 0, B, = 0, 0.^4 = 0, (11) 16Be - IOB4 + j8ff '" = 0, (12) and thenceforward ■ 2; (2j + 6) B^^ - 2j {2j + 3) 5«+, + )8B^ = (13) These equations give B^, B^, ... B^j, ... in succession, in terms of B^, which is so far undetermined. It is obvious, however, from the nature of the problem, that, except for certain special values of h (and therefore of j8), which are such that there is a free oscillation of corresponding type {s = 2) having the speed 2co, the solution must be unique. We shall see, in fact, that unless B^ have a certain definite value the solution above indicated will make the meridian component (u) of the velocity discontinuous at the equator*. The argument is in some respects similar to that of Art. 217. If we denote by N^ the ratio B^j+JBii of consecutive coefficients, we have, from (13), 2i + 3 P 1 2j + 6 2j(2j-\-6)N,' i^..x = ^4^-Kr7.?^.i^, (14) < * In the case of a polar sea bounded by a smaU circle of latitude whose angular radius is ir, the value of B4 is determined by the condition that u^O,or d^/dif =0, at the boundary. 332 Tidal Waves [chap, vm from which it appears that, with increasing j, Nj muBt tend to one or other of the limits and 1. More precisely, unless the limit of N^ be zero, the limiting form of iV^+i will be (2i + 3)/(2i + 6), or 1 - |, approximately. The latter is identical with the limiting form of the ratio of the coefficients of v^ and v^~* in the expansion of (1 — v^y. We infer that, unless £4 have such a value as to make N^ = 0, the terms of the series (10) will become ultimately comparable with those of (1 — v*)*, so that we may write r = i+(l-v«)*M, (15) where i, M are functions of v which do not vanish for v = 1. Near the equator (v = 1) this makes S=^(i-'*)*f=±^ (i«) Hence, by Art. 218 (3), u would change from a certain finite value to an equal but opposite value as we cross the equator. It is therefore essential, for our presept purpose, to choose the value of B^ so that N^ = 0. This is effected by the same method as in Art. 217. Writing (13) in the form N.-MS^ (") 2j + 6 ^'« we see that N^ must be given by the converging continued fraction j3 )3 j3 2i(2i+6)(27+2)(2i + 8) (^j + 4) (2i + 10) 2j + 3 2j + 5 2j + 7 ^ ^^^^ 2^ + 6 2j + 8 2i+10 *''• This holds from J = 2 upwards, but it appears from (12) that it will give also the value of N^ (not hitherto defined), provided we use this symbol for BJH"\ We have then B^ = N^H''\ B, = N,B^, Bs = ^s^e, .... Finally, writing C = f + T* we obtain ^/H'" = v« + N^i^ + N^Ny + N^N^N^i/^ + (19) As in Art. 217, the practical method of conducting the calculation is to assume an approximate value for iV^+i, where j* is a moderately large number, and then to deduce Nj, ^y-i, ... N^, N^ in succession by means of the formula (17). 221] Semi'diurnal Tides 333 The above investigation is taken substantially from the very remarkable paper written by Kelvin* in vindication of Laplace's treatment of the problem, as given in the MAanique Cdeste. In the passage more especially in question, Laplace determines the constant B^ by means of the continued fraction for Ni, without, it must be allowed, giving any adequate justification of the step; and the soundness of this procedure had been disputed by Airyf, and after him by Ferrel}. Laplace, unfortunately, was not in the habit of giving specific references, so that few of his readers appear to have become acquainted with the original presentment! of the kinetic theory, where the solution for the case in question is put in a very convincing, though somewhat different, form. Aiming in the first instance at an approximate solution by means of a finite series, thus : C=B^v* +Bf^u* + . . . + Btt + 2»'****. (20) Laplace remarks || that in ojder to satisfy the differential equation, the coefficients would have to fulfil the conditions 16Be-10JB4+/3JI'"=0, 40JB8 - 28Be + /3B4 = 0, y (21) (2fc -2) (2ifc+4)Btt+, -(2ifc -2) (2ifc + l)Btt+/3B,jt.8=0, -2fc (2ifc +3) Btk+2 +^Btk =0, ^-82*+ 2 =0,-^ as is seen at once by putting Buc^^ =0, Bfjc^t=0, ... in the general relation (13). We have here k + l equations between k constants. The method followed is to determine the constants by means of the first k relations; we thus obtain an exact solution, not of the proposed differential equation (9), but of the equation as modified by the addition of a term /SBu+tv^'^* to the right-hand side. This is equivalent to an alteration of the disturbing force, and if we can obtain a solution such that the required alteration is very small, we may accept it as an approximate solution of the problem in its original form^. Now, taking the first k relations of the system (21) in reverse order, we obtain B^k^t in terms of B^k, thence B^k in terms of B^^i, and so on, until, finally, B4 is expressed in terms of H"'; and it is obvious that if A; be large enough the value of B2k+t, and the consequent adjustment of the disturbing force which is required to make the solution exact, will be very smalL This will be illustrated presently, after Laplace, by a numerical example. The process just given is plainly equivalent to the use of the continued fraction (17) in the manner already explained, starting with J + 1=A;, and Nk=fi/2k{2k+S). The continued fraction, as such, does not, however, make its appearance in the memoir here referred to, but was introduced in the MScanique Cdeste, probably as an after-thought, as a condensed expression of the method of computation originally employed. ♦ Sir W. Thomaon, "On an Alleged Error in Laplace's Theory of the Tides," Phil. Mag. r4), 1. 1. p. 227 (1876) [Papers, t. iv. p. 231]. t "Tides and Waves," ^rt. 111. t "Tidal Researches," U.S. Coast Survey Rep. 1874, p. 154. § "Rechorohes sur quelqnes points da syst^me dn monde," Mim. de VAcad. roy» des Sciences, 1776 [1779] [Oeuvres, t. ix. ppw 187. . .]. II Oeuvres, t. ix. p. 218. The notation has been altered. ^ It is remarkable that this argument is of a kind constantly employed by Airy himself in his researches on waves. 334 Tidal Waves [CHAP, vin The following table gives the numerical values of the coefficients of the several powers of v in the formula (19) for f/fl'", in the cases j3 = 40, 20, 10, 5, 1, which correspond to depths of 7260, 14520, 29040, 58080, 290400 feet, respectively*. The last line gives the value of l^jH"' for v = 1, i.e, the ratio of the amplitude at the equator to its equilibrium-value. At the poles {y = 0), the tide has in all cases the equilibrium-value zero. • j8=40 /3=20 /3 = 10 , 1 /3=5 ^=1 ! y« + 10000 + 1-0000 + 1-0000 + 1-0000 + 1-0000 v^ + 201862 -0-2491 + 6-1915 + 0-7504 + 0-1062 y« + 101164 - 1-4056 + 3-2447 +01666 +00039 y8 -131047 -0-8594 + 0-7234 +00157 +00001 vio - 15-4488 -0-2541 + 00919 +0-0009 1 y" - 7-4581 -00462 +0-0076 v" - 2- 1975 -00058 + 0-0004 y" - 0-4501 -0-0006 y" . - 0-0687 ySO - 0-0082 yM - 0-0008 y»* - 00001 - 7-434 -1-821 + 11-259 + 1-924 + 1110 1 We may use the above results to estimate the closeness of the approximation in each case. For example, when /3=40, Laplace finds 3^^= --000004ff'"; the addition to the disturbing force which is necessary to make the solution exact would then be - -00002^'"y^, and would therefore bear to the actual force the ratio - •00002y'®. It appears from (19) that near the poles, where v is small, the tides are in all cases direct. For sufficiently great depths, jS will be very small, and the formulae (17) and (19) then shew that the tide has everywhere sensibly the equilibrium- value, all the coefficients being small except the first, which is unity. As A is diminished, j8 increases, and the formula (17) shews that each of the ratios Nj will continually increase, except when it changes sign from + to — by passing through the value oo . No singularity in the solution attends this passage of Nj through oo , except in the case of iV^^ , since, as is easily seen, the product N^.^N^ remains finite, and the coefficients in (19) are therefore all finite. But when iV^j =. oo , the expression for { becomes infinite, shewing that the depth has then one of the critical values already referred to. The table above given indicates that for depths of 29040 feet, and upwards, the tides are everywhere direct, but that there is some critical * The first three cases were calculated by Laplace, Ix, ajUe p. 333 ; the last by Kelvin. The numbers relating to the third case have been slightly correcteil, in accordance with the computa- tions of Hough ; see p. 335. 221-222] Hough's Theory 335 depth between 29040 feet and 14520 feet, for which the tide at the equator changes from direct to inverted. The largeness of the second coefficient in the case j3 = 40 indicates that the depth could not be reduced much below 7260 feet before reaching a second critical value. Whenever the equatorial tide is inverted, there must be one or more pairs of nodal circles ($ = 0), symmetrically situated on opposite sides of the equator. In the case of j3 = 40, the position of the nodal circles is pven by V = -95, or fi = 90° ± 18°, approximately*. 222. The dynamical theory of the tides, in the case of an ocean covering the globe, with depth uniform along each parallel of latitude, has been greatly improved and developed by Hough f, who, taking up an abandoned attempt of Laplace, substituted expansions in spherical harmonics for the series of powers of fi (or v). This has the advantage of more rapid convergence, especially, as might be expected, in cases where the influence of the rotation is relatively small; and it also enables us to take account of the mutual attraction of the particles of water, which, as we have seen in the simpler problem of Art. 200, is by no means insignificant. If the surface-elevation ^, and the conventiontd equilibrium tide-height ^ (in which the effect of mutual attraction is not included), be expanded in series of spherical harmonics, thus ^=SC„, ?=Sf„ (1) the complete expression for the disturbing potential will be 2n H- 1 po cf. Art. 200. The series on the right-hand is to be substituted for ^ in the equations of Arts. 214. . . ; this will be allowed for if we write r = S (a„£„ - f„), (2) ^^^'« "'" = i-2;^^„' (^) in modification of the notation of Art. 215 (5) or Art. 218.(4). In the oscillations of the * First Species,' the differential equation may be written If we assume _ f = SC„P„ (jtt). f=Sy«^„(/*). (5) wehave r = S (a„C„ - y„) P„ (jtt) (6) * For a fuller discnssion of these points reference may be made to the original investigation of Laplace, and to Kelyin's papers. t "On the Application of Harmonic Analysis to the Dynamical Theory of the Tides," Phil. Trans. A, t. clxzziz. p. 201, and t. czci. p. 139 (1897). See also Darwin's Papers, t. ii. p. 190. " = -^^(f« + 2lMrip^^«) = 336 Tidal Waves [chap, vm Substituting in (4), and integrating between the limits — 1 and /i, we find (IP r/* S (o„C„ - y„) (1 - /**) ^ + Si8C,{(/« - 1) + (1 - /*«)}] ^ P»dM = 0. (7) Now, by known formulae of zonal harmonics''', and /^/»<^J^ = 2„ + 1 (^»« - ^-i) " 2» + 1 (2n + 3 \ d/tt d/4 / 2n-\\dii dfi 1 <?Pn.H« 2 dP, (2» + 1) (2n + 3) d/[i (2n - 1) (2n + 3) d/j, ^ (2n - 1) (2n + 1) d/x ' ' '^ ' dP Substituting in (7), and equating to zero the coefficient of (1 — jj.*) -^ we find Cn+t — LnCn + TKZ owoIT 1 \ ^«-» = "?>•• (10) (2n + 3) (2n + 5) "■"» " " ' (2n - 3) (2n - 1) "-» j9 where ^" = n (n + 1) ■*■ (2n - .1) (2n + 3) ~ / ^^^^ The relation (10) will hold from n = 1 onwards, provided we put C_i = 0, Co = 0. The further theory is based substantially on the argument of Laplace, given in Art. 221; and the work follows much the same lines as in Arts. 216, 217, 221. In the free oscillations we have y„ = 0, and the admissible values of / are determined by the transcendental equation 1 1 _ J. 5. 7«. 9 9. 11*. 13 - ^« - -T:^ TT^&T = «' (12) 1 1 . 3. 5«. 7 7.9*. 11 - according as the mode is symmetrical or asymmetrical with respect to the equator. Alternative forms of the period equations are given by Hough, ♦ See Todhunter, Functions of Laplace, dfc. c. v. ; Whittaker, Modem Analyais, Art. 117. » , 222-223] Tide9 of Long Period 337 suitable for computation of the higher roots, and it is shewn that close approximations are given by the equations Zf^ » or '« (^ S p^ gh 2 = l + n(n+l)-^ 1 ,.•(14) 4a>* * ' " '" ' *' (V* 2n + 1 />o ^co^a^ (2n - 1) (2n + 3) except for the first two or three values of n. The following table gives the periods (in sidereal time) of the slowest symmetrical oscillation (i.e. the one in which the surface-elevation would vary as Pg (/n) if there were no rotation), corresponding to various depths*. Depth a* Period Period p (feet) 4w* when w=0 40 ' h. m. h. m. 7260 •44155 18 3-5 32 49 20 14520 •62473 15 110 23 12 10 29040 •92506 12 28-6 16 25 6 58080 1-4785 9 52- 1 11 35 The results obtained for the forced oscillations of the * First Species' are very similar to those of Art. 217. The limiting form of the long-period tides when a = shews the following results : 1 a pIp9= •181 pIpo=o ~ 1 1 Pole Equator Pole Equator 40 •140 •426 •154 -455 20 •266 •551 10 •443 •681 •470 •708 5 •628 •796 •651 1 •817 1 The second and third columns give the ratio of the polar and equatorial tides to the respective equilibrium- values f. The numbers in the fourth and fifth columns are repeated from Art. 217. The comparison shews the effect of the mutual gravitation of the water in reducing the amplitude. 223. In the more general case, where symmetry about the axis is not imposed, the surface-elevation { is expanded by Hough in a series of tesseral harmonics of the type Pn (m) e^<'*+'*-^> (1) * The slowest asymmetrical mode has a much longer period. It involves a displacement of the centre of mass of the water, so that a correction would be necessary if the nucleus were free; cf. Art. 199. t The numbers are deduced from Hough's results. The paper referred to includes a discus- sion of many other interesting points, including an examination of cases of varying depth, with numerical illustrations. L. H. 22 338 Tidal WafV€» [oHAP. vm In relation to tidal theory the most important cases are where the disturbing potential is of the form (1), with n = 2 and « = 1 or « = 2. The calculations are necessarily somewhat intricate*^, and it must suffice here to mention a few of the more interesting results, which will indicate how the gaps in the previous investigations have been filled. To understand the nature of the free oscillations, it is best to begin with the case of no rotation (co = 0). As o) is increased, the pairs of numerically equal, but oppositely-signed, values of a which were obtained in- Art. 199 begin to diverge in absolute value, that being the greater which has the same sign with co. The character of the fundamental modes is also gradually altered. These oscillations are distinguished as ' of the First Class.' At the same time certain steady motions which are possible, without change of level, when there is no rotation, are converted into long-period oscillations with change of level, the speeds being initially comparable with CO. The corresponding modes are designated as 'of the Second Class 'f; cf. Art. 206. The following table gives the speeds of those modes of the First Class which are of most importance in relation to the diurnal and semi-diurnal tides, respectively, and the corresponding periods, in sidereal time. The last column repeats the corresponding periods in the case of no rotation, as calculated from the formula (15) of Art. 200. Second Species [* = 1] Third Species [*=2] Period when w=0 h. m. Depth (feet) Period h. m. Period h. m. 7260 14620 29040 58080 1-6337 -0-9834 1-8677 -1-2460 21641 - 1-6170 2-6288 -21611 14 41 24 24 12 61 19 16 11 6 14 60 9 8 11 6 1-3347 -0-6221 1-6133 -0-8922 1-9968 -1-2855 2-5536 - 1-8575 17 69 38 34 14 62 26 54 12 1 18 40 9 24 12 55 1 32 49 1 23 12 1 16 25 1 11 36 * A simplification is made by Love, "Notes on the Dynamical Theory of the Tides," Proc. Lond. Math, Soc. (2), t. zii. p. 309 (1913). He writes ad$ aBm0d<f>* a8mdd<f> add* of. Art. 154 (1). The values of x> ^ ai^ expanded in series of spherical harmonics. f These two classes of oscillations have been already encowitered in the plane problem of Art. 212. 228] Diurnal and Semi'diumal Tides 339 The quickest oscillation of the Second Glass has in each case a period of over a day ; and the periods of the remainder are very much longer. As regards the forced oscillations of the 'Second Species,' Laplace's conclusion that when a = a>, exactly, the diurnal tide vanishes in the case of uniform depth, still holds. The computation for the most important lunar diurnal tide, for which a/co = '92700, shews that with such depths as we have considered the tides are small compared with the equilibrium heights, and are in the main inverted. Of the forced oscillations of the 'Third Species,' we may note first the case of the solar semi-diurnal tide, for which a = 2ai with sufficient accuracy. For the four depths given in our tables, the ratio of the dynamical tide-height to the conventional equilibrium tide-height at the equator is found to be + 7-9548, - 1-5016, - 23487, -f 2-1389, respectively. **The very large coefficients which appear when hg/4:a)^d^ = -^ indicate that for this depth there is a period of free oscillation of semi-diurnal type whose period differs but slightly from half-a-day. On reference to the tables ... it will be seen that we have, in fact, evaluated this period as 12 hours 1 minute, while for the case hg/4o)^a^ == tV ^® have found a period of 12 hours 5 minutes*. We see then that though, when the period of forced oscillation differs from that of one of the types of free oscillation by as little as one minute, the forced tide may be nearly 250 times as great as the corresponding equilibrium tide, a difference of 5 minutes between these periods will be sufficient to reduce the tide to less than ten times the corresponding equilibrium tide. It seems then that the tides will not tend to become excessively large unless there is very close agreement with the period of one of the free oscillations. "The critical depths for which the forced tides here treated of become infinite are those for which a period of free oscillation coincides exactly with 12 hours. They may be ascertained by putting [a = 2co] in the period- equation for the free oscillations and treating this equation as an equation for the determination of A The two largest roots are. . . , and the corre- sponding critical depths are about 29,182 feet and 7375 feet "It will be seen that in three cases out of the four here considered the effect of the mutual gravitation of the waters is to increase the ratio of the tide to the equilibrium tide [cf. Art. 221]. In two of the cases the sign is also re- versed. This of course results from the fact that whereas when [p/pi = 0* 18093] one of the periods of free oscillation is rather greater than 12 hours, when [pIpi = 0] the corresponding period will be less than 12 hoursf." * [Belonging to a mode which comee next in sequence to the one haying a period of 17 h. 59 m.] t Hough, Phil. Trans, A, t. ozoi. pp. 178, 179. 22—2 340 Tidal Waves [chap, vni Hough has also computed the lunar semi-diurnal tides for which ^ = 0-96350. For the four depths aforesaid the ratios of the equatorial tide-heights to their equilibrium- values are found to be - 2-4187, - 1-8000, + 11-0725, + 1-9225, respectively. ''On comparison of these numbers with those obtained for the solar tides. . . , we see that for a depth of 7260 feet the solar tides will be direct while the lunar tides will be inverted, the opposite being the case when the depth is 29,040 feet. This is of course due to the fact that in each of these cases there is a period of free oscillation intermediate between twelve solar (or, more strictly, sidereal) hours and twelve lunar hours. The critical depths for which the limar tides become infinite are found to be 26,044 feet and 6448 feet. ** Consequently this phenomenon will occur if the depth of the ocean be between 29,182 feet and 26,044 feet, or between 7375 feet and 6448 feet. An important consequence would be that for depths lying between these limits the usual phenomena of spring and neap tides would be reversed, the higher tides occurring when the moon is in quadrature, and the lower at new and full moon*." The most recent contribution to the dynamical theory consists of two papers by Goldsbroughf, who has discussed the tides in an ocean of uniform depth limited by one or two parallels of latitude. In the case of a polar basin of angular radius 30^, for instance, he finds that for such depths as have been considered in Arts. 217, 221 the long-period tides and the semi- diurnal tides do not deviate very widely from the values given by the equi- librium theory, when this is corrected as explained in the Appendix. The case is however very different with the diurnal tides, which vary considerably with the size of the basin and the depth, and are as a rule considerable, whereas we have seen that in a uniform ocean covering the globe they are negligible. In the case of an equatorial belt, the long-period tides again approximate to the equilibrium- values, whilst the diurnal and semi-diurnal deviate widely, to an extent which varies considerably with the position of the boundaries. The more difficult case of an ocean bounded by two meridians has not yet been investigated, except on the supposition that the angular velocity of * Hough, lx,j where reference is made to Kelvin's Popular Lectures and Addresses, London, 1894, t. ii. p. 22 (1868). t "The Dynamical Theory of the Tides in a Polar Basin,** Proc Lond. Math. 8oc. (2), t. ziv. p. 31 (1913); "The Dynamical Theory of the Tides in a Zonal Ocean,** ibid. p. 207 (1914). 223-224] Lag of the Tides 341 rotation is small compaied with the speed of the free oscillations*. The plane problem of a circular sector would appear to be somewhat simpler, but has not been solved. 224. It is not easy to estimate, in any but the most general way, the extent to which the foregoing conclusions of the dynamical theory would have to be modified if account could be taken of the actual configuration of the ocean, with its irregular boundaries and irregular variation of depthf . One or two points may however be noticed. In the first place, the formulae (17) of Art. 206 would lead us to expect for any given tide a phase-difference, variable from place to place, between the tide-height and the disturbing force:|:. Thus, in the case of the lunar semi-diurnal tides, for example, high- water or low-water need not synchronize with the transit of the moon or anti-moon across the meridian. More precisely, in the case of a disturbing force of given type for which the equilibrium tide-height at a particular place would be f SB a cos cr^, (1) the dynamical tide-height will be J = ii cos (cr« - €), (2) where the ratio Aja^ and the phase-diSerence €, will be functions of the speed a. Again, consider the superposition of two oscillations of the same type but of slightly different speeds, e.g. the lunar and solar semi-diurnal tides. If the origin of t be taken at a syzygy, we have f = a cos (jt-\- a' cos a% (3) and C^ Acqs {at — e) -\- A' cos {at — e') (4) This may be written ^= {A-\- A' cos^) cos {ai — e) -\- A' sin^ sin {ai — e), (5) where <f> =^ {a - (/)t - e -{- e (6) If the first term in the second member of (4) represents the lunar, and the second the solar tide, we shall have a < a, and A > A'. If we write A + A' co8(f> = C cos a, il' sin ^ = C sin a, (7) we get f = C cos (<rf — € — a), (8) where C = {A^ + 2 A A' cos <f> -f- A'^)^ , a = tan-i . f ' !f "^ i . • - (9) A. ^~ A. COS u) * Rsyleigh, PnK. Boy. Boc A, t. Ixzxii. p. 448 (1909) [Pa/pen, t. ▼. p. 497]. t As to the general mathematioal problem reference may be made to Poinoar6, **Snr I'^ni- libre et les mouyements dee men,*' LumviUe (5), t. ii. pp. 57, 217 (1896), and to his Lefona de m^canique UUaU, t. iiL % This U illustrated by the canal problem of Art. 184. 342 Tidal Waves [chap, vra This may be described as a simple-harmonic oscillation of slowly varying amplitude and phase. The amplitude ranges between the limits A ± A\ whilst a may be supposed to lie always between ± \7t. The 'speed' must also be regarded as variable, viz. we find da __ o-^» 4- (o- + a) A A' cos<^ + aA'^ "" dt A*'h2AA'coB<f> + A'^ ^^^^ This ranges between Aa + AW , Aa-AV ,,,,^ A-TA'- ^^* -Z3^' (^^) The above is the well-known explanation of the phenomena of the spring- and neap- tides t; but we are now concerned further with the question of phase. On the equilibrium theory, the maxima of the amplitude C would occur whenever {a — a)t = 2mr, where n is integral. On the dynamical theory the corresponding times of maximum are given by (a - a) <-(€'-€) = 2n7r, i.e. the dynamical maxima follow the statical by an interval | (€' - e)/(a' - a). If the difference between a and a were infinitesimal, this would be equal to defda. The fact that the time of high-water, even at syzygy, may follow or precede the transit of the moon or anti-moon by an interval of several hours is well known §. The interval, when reckoned as a retardation, is, moreover, usually greater for the solar than for the lunar semi-diurnal tide, with the result that the spring-tides are in many places highest a day or two after the corresponding syzygy. The latter circumstance has been ascribed || to the operation of Tidal Friction (for which see Chapter xi.), but it is evident that the phase-difierences which are incidental to a complete dynamical theory, even in the absence of friction, cannot be ignored in this connection. There is reason to believe that they are, indeed, far more important than those due to the latter cause. Lastly, it was shewn 'in Arts. 206, 217 that the long-period tides may deviate very considerably from the values given by the equihbrium theory, * Helmholtz, Lfhre von den Tonempfindungen (2« Aufi.), Braunaohweig, 1870, p. 622. t Cf . Thomaon and Tait. Art. 60. } This inteiral may of oonrse be negative. § The values of the retardations (which we have denoted by e) for the various tidal com- ponents, at a number of ports, are given by Baird and Bandn, "Results of the Harmonic Analysis of Tidal Observations," Proc, B. S, t. xxxix. p. 135 (1885), and Darwin, ♦^Second Series of Results. . . ," Proc. JR. S, t. xlv. p. 666 (1889). II Airy, "Tides and Waves," Art. 459 224-225] Stability of the Ocean 343 0¥H[ng to the possibility of certain steady motions in the absence of disturbance. It has been pointed out by Bayleigh* that these steady motions may be impossible in certain cases where the. ocean is limited by perpendicular barriers. Referring to Art. 214 (6), it appears that if the depth h be imiform, ^ must (in the steady motion) be a function of the co-latitude 6 only, and therefore by (4) of the same Art., the eastward velocity v must be uniform along each parallel of latitude. This is inconsistent with the existence of a perpendicular barrier extending along a meridian. The objection would not necessarily apply to the case of a sea shelving gradually from the central parts to the edgef . 225. We may complete the investigation of Art. 200 by a brief notice of the question of the stability of the ocean, in the case of rotation. It has been shewn in Art. 205 that the condition of secular stability is that V — Tq should be a minimum in the equilibrium configuration. If we neglect the mutual attraction of the elevated water, the application to the present problepi is very simple. The excess of the quantity F — Tq ^^er its undisturbed value is evidently -\a}^w^)dz'^dS, (1) where ^ denotes the potential of the earth's, attraction, 85 is an element of the oceanic surface, and the rest of the notation is as before. Since "9 — Joi^td* is constant over the undisturbed level {z = 0), its value at a small altitude z may be taken to he gz -\- const., where, as in Art. 213, =[ I (^ - W^^) (2) Since SS^dS = 0, on account of the constancy of volume, we find from (1) that the increment of F — To is iM'dS (3) This is essentially positive, and the equilibrium is therefore secularly stable :|:. It is to be noticed that this proof does not involve any restriction as to the depth of the fluid, or as to smallness of the ellipticity, or even as to symmetry of the undisturbed surface with respect to the axis of rotation. If we wish to take into account the mutual attraction of the water, the problem can only be solved without difficulty when the undisturbed surface is nearly spherical, and we neglect the variation of g. The question (as to secular stability) is then exactly the same as in the case of no rotation. ♦ "Note on the Theory of the Fortnightly Tide," PhU, Mag, (6), t. v. p. 136 (1903) [Papers^ t. lY. p. 84]. t The theory of the limiting forms of long-period tides in oceans of various types is discussed by Proudman, Proc. Lond. Math. Soc. (2), t. xiii. p. 273 (1913). t Cf . Laplace. Mdcaniqt^e OiUate, Livre 4me, Arts. 13, 14. 344 Tidal Waves [chap, vin The calculation for this case will find an appropriate place in the next chapter. The result, as we might anticipate from Art. 200, is that the necessary and sufficient condition o£ stability of the ocean is that its density should be less than the mean density of the earth*. 226. This is perhaps the most suitable occasion for a few additional remarks on the general question of stability of dynamical systems. We have in the main followed the ordinary usage which pronounces a state of equilibrium, or of steady motion, to be stable or unstable according to the character of the solution of the approximate equations of disturbed motion. If the solution consists of series of terms of the type Ce***, where all the values of A are pure imaginary (i.e. of the form ia), the undisturbed state is usually reckoned as stable ; whilst if any of the A's are real, it is accounted unstable. In the case of disturbed equilibrium^ this leads algebraically to the usual criterion of a minimum value of F as a necessary and sufficient condition of stability. It has in recent times been questioned whether this conclusion is, from a practical point of view, altogether warranted. It is pointed out that since Lagrange's equations become less and less accurate as the deviation from the equilibrium configuration increases, it is a matter for examination how far rigorous conclusions as to the ultimate extent of the deviation can be drawn from themf. The argument of Dirichlet, which establishes that the occurrence of a minimum value of F is a sufficient condition of stability, in any practical sense, has already been referred to. No such simple proof is available to shew without qualification that this condition is necessary. If, however, we recognize the existence of dissipative forces, which are called into play by any motion whatever of the system, the conclusion can be drawn as in Art. 205. A little consideration will shew that a good deal of the obscurity which attaches to the question arises from the want of a sufficiently precise mathematical definition of what is meant by 'stability.' The difficulty is encountered in an aggravated form when we pass to the question of stability of motion. The various definitions which have been propoimded by difierent writers are examined critically by Klein and Sommerfeld in their book on the theory of the top J. Rejecting previous definitions, they base their criterion on the character of the changes produced in the jxUh of the system by small arbitrary disturbing impulses. If the undisturbed path be the limiting form of the disturbed path when the impulses are indefinitely ♦ Cf. Laplace, M4canique Cdeste, Livre 4™®, Arts. 13, 14. f See papers by Liapounofif and Hadamard, Lumville (5), t. iii. (1897). } Ueber die Theorie des Kreisds. Leipzig, 1897 . . . , p. 342. \ 225-226] Dynamical Stability 345 diminished, it is said to be stable, but not otherwise. For instance, the vertical fall of a particle under gravity is reckoned as stable, although for a given impulsive disturbance, however small, the deviation of the particle's position at any time t from the position which it occupied in the original motion increases indefinitely with t. Even this criterion, as the writers referred to themselves recognize, is not free from ambiguity imless the phrase ^limiting form,' as applied to a path, be strictly defined. It appears moreover that a definition which is analytically precise may not in all cases be easy to reconcile with geometrical prepossessions*. The foregoing considerations have reference, of course, to the question of 'ordinary' stability. The more important theory of 'secular' stability {Alt, 206) is not afiected. We shall meet with the criterion for this, under a somewhat modified form, at a later stage in our subject f* * Some good illustrations are famished by Particle Dpiffcmics. Thus a particle moving in a circle about a centre of force ▼ar3ang inversely as the cube of the distance will if slightly disturbed either fall into the centre, or recede to infinity, after describing in either case a spiral with an infinite number of convolutions. Each of these spirals has, analytically, the circle as its * limiting form,' although the motio>n in the latter is most naturally described as unstable. Of. Korteweg, Wiener Ber. May 20, 1886. A narrower definition has been given by Love, and applied by Bromwich to several dynamical and hydrodynamioal problems; see Proc. Land, Math. Soc t. zxxiii. p. 325 (1901). t This summary is taken substantially from the Art. "Dynamics, Analytical,*' in Encyc. Brit, 10th ed., t. uvii. p. 566 (1902), and 11th ed., t. viu. p. 756 (1910). 4 * I I I ^ APPENDIX TO CHAPTER VIII ON TIDE-GENERATING FORCES a. If, in the annexed figure, and C be the centres of the earth and of the disturbing body (say the moon), the potential of the moon's attraction at a point P near the earth's surface will be -yJUfCP, %«kcre M denotes the moon's mass, and y the gravitation- constant. If we put OC=D, OP=r, «fid denote the moon's (geocentric) zenith-distance at P, viz. the angle POC, by 3, this potential is equal to yM (Z)«-2rDcos3 + r«)i* We require, however, not the absolute accelerative effect at P. buiVthe acceleration relative to the earth. Now the moon produces in the whole mass ol the earth an acceleration yM/D^* parallel to OC, and the potential of a uniform field o^ force of thip intensity is evidently y3f D« . r cos S, Subtracting this from the former result we get, for the potential of the relative V^^^^^^^^ atP, ^ + ^ . r cos ^ o= - (1) {D^ -2rD COS S+r^)i ^* This function Q is identical with the * disturbing-function' of planetary theory. Expanding in powers of r/D, which is in our case a small quantity, and retainin]| only the most importeuit term, we find o=*^ i ^(i-cos«5). Considered as a function of the position of P, this is a zonal harmonic of the 8< degree, with OC as axis. 5) md • The effect of this is to produce a monthly inequality in the motion of the earth's cenV'^ about the sun. The amplitude of the inequality in radius vector is about 3000 miles; that I of the inequality in longitude is about 7"; see Laplace, M4canique COesie, livre &^, Art. 30, apd Livre 13"»% Art. 10. APR] Equilibrium Theory 847 The reader will easily verify that, to the order of approximation adopted, Q is equal to the joint potential of two masses, each equal to ^if, placed, one at C, and the other at a point C in CO produced such that OC =0C*. b. In the 'equilibrium-theory' of the tides it is assumed that the free surface takes at each instant the equilibrium-form which might be maintained if the disturbing body were to retain unchanged its actual position relative to the rotating earth. In other words, the free surface is assumed to be a level-surface under the combined action of gravity, of centrifugal force, and of the disturbing force. The equation to this level- surface is * -^•oj* +Q =const (3) where <i> is the angular velocity of the rotation, w denotes the distance of any point from the earth's axis, and ^ is the potential of the earth's attraction. If we use square brackets [ ] to distinguish the values of the enclosed quantities at the undisturbed level, and denote by f the elevation of the water above this level due to the disturbing potential 12, the above equation is equivalent to [* - Ja)«w«] +r^(* -io)"w«)|f +12 = const., (4) approximately, where d/dz is used to indicate a space-differentiation along the normal outwards. The first term is of course constant, and we therefore have C--^+C (6) where, as in Art. 213, ^=r^(* -i»*ar«)1 (6) Evidently, g denotes the value of 'apparent gravity'; it will of course vary more or less with the position of P on the earth's surface. It is usual, however, in the theory of the tides, to ignore the slight variations in the value of g, and the effect of the eUipticity of the undisturbed level on the surface- value of a. Putting, then, r=:a, g=yE/a\ where E denotes the earth's mass, and a the mean radius of the surface, we have, from (2) and (6), f=jy(cos»^-J)+C (7) where ^=|.|. («)•.., (8) as in Art. 180. Hence the equilibrium-form of the free surface is a harmonic spheroid of the second order, of the zonal type, whose axis passes through the disturbing body. C. Owing to the diurnal rotation, and also to the orbital motion of the disturbing body, the position of the tidal spheroid relative to the earth is continually changing, so that the level of the water at any particular place will continually rise and fall. To anal3rse the character of these changes, let be the co-latitude, and <^ the longitude, measured eastward from some fixed meridian, of any place P, and let Abe the north-polar- distance, and a the hour-angle west of the same meridian, of the disturbing body. We have, then, cos S =co8 A cos B +sin A sin cos (a +<^), (9) * Thomson and Tait, Art. 804. These two fictitious bodies are designated as 'moon' and 'anti-moon,' respectively. 348 On Tide-Grcnerating Forces [chap, vin and thenoe, by (7), C=f£r(ooe«A-i)(oo8«^-i) + ^ JJ sin 2a sin 2^ COS (a + 0) +i£r8m« Asin* ^ ooe 2 (a +<^) +C (10) Each of these terms may be regarded as representing a partial tide, and the results superposed. Thus, the first term is a zonal harmonic of the second order, and gives a tidal spheroid symmetrical with respect to the earth's axis, having as nodal lines the parallels for which cos* B=i,otO= 90° ±, 35° 16^ The amount of the tidal elevation in any particular latitude varies as cos' A - J . In the case of the moon the chief fluctuation in this quantity has a period of about a fortnight; we have here the origin of the * lunar fortnightly' or *declinationar tide. When the sun is the disturbing body, we have a 'solar semi-annual' tide. It is to be noticed that the mean value of cos' A - ^ with respect to the time is not zero, so that the inclination of the orbit of the disturbing body to the equator involves as a consequence a permanent change of mean level. Of. Art. 183. The second term in (10) is a spherical harmonic of the type obtained by putting n =2, « = 1 in Art. 86 (7). The corresponding tidal spheroid has as nodal lines the meridian which is distant 90° from that of the disturbing body, and the equator. The disturbance of level is greatest in the meridian of the disturbing body, at distances of 45° N. and S. of the equator. The osciUation at any one place goes through its period with the hour- angle, a, i.e. in a lunar or solar day. The amplitude is, however, not constant, but varies slowly with A, changing sign when the disturbing body crosses the equator. This term accounts for the lunar and solar *diumar tides. The third term is a sectorial harmonic (n=2, «=2), and gives a tidal spheroid having as nodal lines the meridians which are distant 46° E. and W. from that of the disturbing body. The oscillation at any place goes through its period with 2a, t.e. in half a (lunar or solar) day, and the amplitude varies as sin' A, being greatest when the disturbing body is on the equator. We have here the origin of the lunar and solar * semi-diurnal' tides. The 'constant' O is to be determined by the consideration that, on account of the invariability of volume, we must have /Jf(Mf=0 (11) where the integration extends over the surface of the ocean. If the ocean cover the whole earth we have C =0, by the general property of spherical surface-harmonics quoted in Art. 87. It appears from (7) that the greatest elevation above the undisturbed level is then at the points .9=0, ^ = 180°, t.e. at the points where the disturbing body is in the zenith or nadir, and the amount of this elevation is f if. The greatest depression is at places where ^ =90°, i.e. the disturbing body is on the horizon, and is f JST. The greatest possible range \a therefore equal to H, In the case of a limited ocean, C does not vanish, but has at each instant a definite value depending on the position of the disturbing body relative to the earth. This value may be easily written down from equations (10) and (11); it is a sum of spherical harmonic functions of A, a, of the second order, with constant coefficients in the form of surface-integrals whose values depend on the distribution of land and water over the globe. The changes in the value of C, due to relative motion of the disturbing body, give a general rise and fall of the free surface, with (in the case of the moon) fortnightly, diurnal, and semi-diurnal periods. This 'correction to the equilibrium-theory' as usuaUy APP.] Harmonic Analysis 349 presented, was first fully investigated by Thomson and Tait*. The necessity for a oorreotion of the kind, in the case of a limited sea, had however been recognized by D. BemouUif. The correction has an influence on the time of high water, which is no longer synchronous with the maximum of the disturbing potential. The interval, moreover, by which high water is accelerated or retarded differs from place to place:(. d. We have up to this point neglected the mutual attraction of the particles of the water. To take this into account, we must add to the disturbing potential Q the gravitation-potential of the elevated water. In the case of an ocean covering. the earth, the correction can be easily applied, as in Art. 200. If we put n=2 in the formulae of that Art., the addition to the value of Q is -^p/po'gCi ^^^ ^® thence find without difficulty ^=n«;<°"*'-*^ <''> It appears that all the tides are increasedf in the ratio (1 -fp/pu)~^* ^ ^^ assume p/po = *18, this ratio is 1-12. e. So much for the equilibrium-theory. For the purposes of the kinetic theory of Arts. 21^224, it is necessary to suppose the value (10) of ^ to be expanded in a series of simple-harmonic functions of the time. The actual expansion, taking account of the variations of A and a, and of the distance 2> of the disturbing body (which enters into the value of H), is a somewhat complicated problem of Physical Astronomy, into which we do not enterf . Disregarding the constant C, which disappears in the dynamical equations (1) of Art. 215, the constancy of volume being now secured by the equation of continuity (2), it is easily seen that the terms in question will be of three distinct types. First, we have the tides of long period, for which C=H' (coe« e -J) . cos {(Tt +€) (13) The most importeutit tides of this class are the * lunar fortnightly' for which, in degrees per mean solar hour, o- = 1°'098, and the *solar-annuar for which 0-= 0^*082. Secondly, we have the diurnal tides, for which C=H" BiaBoo&e.co6{(rt-\-<l) +«), (14) where <r differs but little from the angular velocity o> of the earth's rotation. These include the *lunar diurnal' (<r = 13**-W3), the 'solar diurnal' (a = 14°-969), and the Muni- solar diurnal' (cr =» = 15^-041). * Natural Philosophy, Art. 808; see also Darwin, "On the Correction to the Equilibrium Theory of the Tides for the Continents," Proc. Roy. 80c. April 1, 1886 [Papers, t. i. p. 328]. It appears as the result of a numerical calculation by Prof. H. H. Turner, appended to this paper, that with the actual distribution of land and water the correction is of little importance. t Traits aur U Flux ei Reflux de la Mer, c. xL (1740). This essay, as well as the one by Maclaurin cited on p. 300, and another on the same subject by Euler, is reprinted in Le Seur and Jacquier's edition of Newton's Principia, X Thomson and Tait, Art. 810. The point is illustrated by the formula (3) of Art. 184 supra. § Reference may be made to lAplaoe, Micaniqite Celeste, Livre 13"**, Art. 2, and to Darwin*s Papers, t. i. 850 On Tide-Generating Forces [chap, vm Lastly, we have the semi-diurnal tides, for which f =fr'" sin* e . oosM +2<^ +*), . . (16)* where <r differs but little from 2<o. These include the * lunar semi-diurnal' (<r =28^-984), the * solar semi-diurnal* (<r =30°), afid the *luni-solar semi-diurnal* (tr =2<o =30^-082). For a complete enumeration of the more important partial tides, and for the values of the coefficients H\ H"y H"' in the several cases, we must refer to the investigations of Darwin, already cited. In the Harmonic Analysis of Tidal Observations, which is the special object of these investigations, the only result of dynamical theory which is made use of is the general principle that the tidal elevation at any place must be equal to the sum of a series of simple-harmonic functions of the time, whose periods are the same as those of the several terms in the development of the disturbing potential, and are therefore known d 'priori. The amplitudes and phases of the various partial tides, for any particular port, are then determined by comparison with tidal observations extending over a sufficiently long period f. We thus obtain a practically complete expression which can be used for the systematic prediction of the tides at the port in question. f. One point o( special interest in the Harmonic Analysis is the determination of the long-period tides. It has been already stated that under the influence of dissipative forces these must tend to approximate more or less closely to their equilibrium values. Unfortunately, the only long-period tide, whose coefficient can be inferred with any certainty from the observations, is the lunar fortnightly, and it is at least doubtful whether the dissipative forces are sufficient to produce in this case any great effect in the direction indicated. Hence the observed fact that the fortnightly tide has less than its equilibrium value does not entitle us to make any inference as to elastic yielding of the solid body of the earth to the tidal distorting forces exerted by the moon %. * It is evident that over a small area, near the poles, which may be treated as sensibly plane, the formulae (14) and (15) make ^« r COB (<r< + 0-1-6), and f a r * cos (<ri -i- 2^ -|- e), respectively, where r, w are plane polar co-ordinates. These forms have been used by anticipation in Arts. 211, 212. f It is of interest to note, in connection with Art. 187, that the tide-gauges, being situated in relatively shallow water, are sensibly affected by certain tides of the second order, which there- fore have to be taken account of in the general scheme of Harmonic Analysis. } Darwin, Uc, ante p. 323. See, however, the paper by Bayleigh cited on p. 343 ante. CHAPTER IX SURFACE WAVES 227. We have now to investigate, as far as possible, the laws of wave- motion in liquids when the vertical acceleration is no longer neglected. The most important case not covered by the preceding theory is that of waves on relatively deep water, where, as will be seen, the agitation rapidly diminishes in amplitude as we pass downwards from the surface; but it will be imderstood that there is a continuous transition to the state of things investigated in the preceding chapter, where the horizontal motion of the fluid was sensibly the same from top to bottom. We begin with the oscillations of a horizontal sheet of water, and we will confine ourselves in the first instance to cases where the motion is in two dimensions, of which one {x) is horizontal, and the other (y) vertical. The elevations and depressions of the free surface wiU then present the appearance of a series of parallel straight ridges and furrows, perpendicular to the plane xy. The motion, being assumed to have been generated originally from rest by the action of ordinary forces, will necessarily be irrotational, and the velocity-potential <f> will satisfy the equation al + a? = *' (1) with the condition ^ "^^ (2) at a fixed boundary. To find the condition which must be satisfied at the free surface {p r= const.), let the origin be taken at the undisturbed level, and let Oy be drawn vertically upwards. The motion bdng assumed to be infinitely small, we find, putting Cl=^gy 'm the formula (4) of Art. 20, and neglecting the square of the velocity (g), M-»»+^('). (3) 352 Surfa^^e Waves [chap, ix Hence if t\ denote the elevation of the surface at time i above the point {Xy 0)» we shall have, since the pressure there is uniform, -^[ti... '« provided the function F (^), and the additive constant, be supposed merged in the value of 3<^/3^ Subject to an error of the order already neglected, this may be written -HIL '^' Since the normal to the free surface makes an infinitely small angle (drj/dx) with the vertical, the condition that the normal component of the fluid velocity at the free surface must be equal to the normal velocity of the surface itself gives, with sufficient approximation, l-[IL '^' This is in fact what the general surface condition (Art. 9 (3)) becomes, if we put F {x, y^ z, t) = y — t), and neglect small quantities of the second order. Eliminating rj between (5) and (6), we obtain the condition dt* ^dy ' ^ ' to be satisfied when y = 0. This is equivalent to Dp/Dt = 0. In the case of simple-harmonic motion, the time-factor being e*<'*+*^, this condition becomes <^<f> = 9^ (8) 228. Let us apply this to the free oscillations of a sheet of water, or a straight canal, of uniform depth h, and let us suppose for the present that there are no limits to the fluid in the direction of x, the fixed boundaries, if any, being vertical planes parallel to xy. Since the conditions are uniform in respect to x, the simplest supposition we can make is that ^ is a simple-harmonic function of x ; the most general case consistent with the above assumptions can be derived from this by superposition, in virtue of Fourier's Theorem. We assume then <f> = P cos Jcx, e*<'*+'>, , (1) where P is a function of y only. The equation (1) of Art. 227 gives ^-**^ = 0. (2) whence P = Ae^^ + Be-^^ (3) 227-228] Standing Waves 353 The condition of no vertical motion at the bottom is 3^/3y = for y = — A, whence ile-** = Be**, = JC, say. This leads to <f> = C cosh k{y -^ h) cos kx . e*<*'*+*^ (4) The valuie of a is then determined by Art. 227 (8), which gives or* = gk tanh kh (5) Substituting from (4) in Art. 227 (5), we find iaC T] = — cosh kh cos kx . e*^'*+'^ (6) 9 or, writing a = . cosh M, and retaining only the real part of the expression, 7) = a cos fee . sin (o^ + €) (7) This represents a system of 'standing waves,' of wave-length A = 27r/i, and vertical amplitude a. The relation between the period (27r/or) and the wave-length is given by (5). Some numerical examples of this dependence are given on p. 357. In terms of a we have , qa cosh Jk (v + A) , / . v ^ = -a coshM ^ cos to ■ coa (a< + e). (8) and it is easily seen from Art. 62 that the corresponding value of the stream- function is , flra sinh i (y + A) . , /^ . . //^v 'f'a coshM ' ^*^-°"°('^ + ^) (^> If X, y be the co-ordinates of a particle relative to its mean position (x, y), we have it" dx' dJt'' dy' ^^' if we neglect the differences between the component velocities at the points (x, y) and (x -\-x,y + y), as being small quantities of the second order. Sub- • stituting from (8), and integrating with respect to t, we find cosh k(y + h) , J • /^ , \ X = — a . ^j, — sm kx . sm (ctt + e), smhM . .--. sinh k(y + h) , . , ^ . f y = a . , TT — ' cos kx . sm ((ji -f- c), smhM ^ 'V where a slight reduction has been effected by means of (5). The motion of each particle is rectilinear, and simple-harmonic, the direction of motion varjring from vertical, beneath the crests and hollows {kx = mrr), to horizontal, beneath the nodes (kx = {m + I) n). As we pas6 downwards from the surface L. H. 23 354 Surface Waves [chap. IX to the bottom the amplitude of the vertical motion diminishes from a cos he to 0, whilst that of the horizontal motion diminishes in the ratio cosh kh:\. When the wave-length is very small compared with the depth, kh is large, and therefore tanh kh= 1*. The formulae (11) then reduce to X = — ae^y sin kx . sin (at + c), y = ae^^ cos kx , sin {<ft + e), . . (12) with a^ = gk, .^ (13) The motion now diminishes rapidly from the surface downwards; thus at a depth of a wave-length the diminution of amplitude is in the ratio c"*' or 1/535. The forms of the lines of (oscillatory) motion (iff = const.), for this case, are shewn in the annexed figure. In the above investigation the fluid is supposed to extend to infinity in the direction of x, and there is consequently no restriction to the value of k. The formulae also give, however, the longitudinal oscillations in a canal of finite length, provided k have the proper values. If the fluid be bounded by the vertical planes x = 0, x^l (say), the condition 3^/3x = is satisfied at both ends provided sin AZ = 0, or kl = mw; where m = 1, 2, 3, The wave-lengths of the normal modes are therefore given by the formula A = 2l/m. Cf. Art. 178. 229. The investigation of the preceding Art. relates to the case of ^standing' waves; it naturally claimed the first place, as a straightforward application of the usual method of treating the free oscillations of a system about a state of equilibrium. In the case, however, of a sheet of water, or a canal, of uniform depth, extending horizontally to infinity in both directions, we can, by super- position of two systems of standing waves of the same wave-length, obtain a system of progressive waves which advance unchanged with constant velocity. For this, it is necessary that the crests and troughs of one component system should coincide (horizontally) with the nodes of the other, that the amplitudes of the two systems should be equal, and that their phases should differ by a quarter-period. * This case may of course be more easily investigated independently. 228^-229] Progressive Waves 355 Thus if we put V^Vi^Vi* (1) where 7ji=^ a^kx cos <ft, 172 '= ^ ^^^ kx mi at^ (2) we get 7) = a sin {kx ± at), * (3) which represents an infinite train of waves travelling in the negative or positive direction of a;, respectively, with the velocity c given by c = ?=(|tanhM)*, (4) where the value of a has been substituted from Art. 228 (5). In terms of the wave-length (A) we have c -(£-*x)* <») When the wave-length is anything less than double the depth, we have tanh M = 1, sensibly, and therefore* c -©*-©* («) On the other hand when A is moderately large compared with h we have tanh kh » M, nearly, so that the velocity is independent of the wave-length, being given by c = {9h)\ 0) as in Art. 170. This result is here obtained on the assumption that the wave-profile is a curve of sines, but Fourier's Theorem shews that the restriction is now to a great extent unnecessary. It appears, on tracing the curve y « (tanh x)/x, or from the numerical table to be given presently, that for a given depth h the wave- velocity increases constantly with the wave-length, from zero to the asymptotic value (7). Let us now fix our attention, for definiteness, on a train of simple-harmonic waves travelling in the positive direction, i.e. we take the lower sign in (1) and (3). It appears, on comparison with Art. 228 (7), that the value of ^1 is deduced by putting € = Jtt, and subtracting Jtt from the value of fccf, and that of 7j2 by putting c = 0, simply. This proves a statement made above as to the relation between the component systems of standing waves, and also enables us to write down at once the proper modifications of the remaining formulae of the preceding Art. * Green, "Note on the Motion of Waves in Canals," Camb. Trans, t. vii. (1839) [Papers, p. 279]. t This is merely equivalent to a change of the origin from which x is measured. 2a— 2 356 , Surface Waves [chap, ix TIuis, we find, for the component displacements of a particle, cosh A (y + A) ,, ^, ^ sinh A (y + A) . ,, ^. (8) This shews that the motion of each particle is elliptic-harmonic, the period (27r/(7, = X/c) being that in which the disturbance travels over a wave- length. The semi-axes, horizontal and vertical, of the elliptic orbits are ^ cosh kjy-jrh) ^^^ ^ sinh k{y-}-h) sinhM sinhA;A ' respectively. These both diminish from the surface to the bottom (y = — h), where the latter vanishes. The distance between the foci is the same for all the ellipses, being equal to acosechM. It easily appears, on comparison of (8) with (3), that a surface-particle is moving in the direction of wave- propagation when it is at a crest, and in the opposite direction when it is in a trough*. When the depth exceeds half a wave-length, c~** is very small, and the formulae (8) reduce to X = oe**' cos {Jcx — at), y = ae^^ sin {kx — at), (9) so that each particle describes, a circle, with constant angular velocity a, = (2Trgr/A)'t- The radii of these circles are given by the formula ae^^, and therefore diminish rapidly downwards. In the first table on the next page, the second colunm gives the values of sech kh corresponding to various values of the ratio A/A. This quantity measures the ratio of the horizontal motion at the bottom to that at the surface. The third column gives the ratio of the vertical to the horizontal diameter of the elliptic orbit of a surface-particle. The fourth and fifth columns give the ratios of the wave-velocity to that of waves of the same length on water of infinite depth, and to that of *long' waves on water of the actual depth, respectively. The tables of absolute values of periods and wave- velocities, which are also given on p. 357, are abridged from Airy's treatise J. The value of g adopted by him is 32* 16 ft./sec.«. The possibility of progressive waves advancing with imchanged form is limited, theoretically, to the case of uniform depth; but the numerical ♦ The results of Arts. 228, 229, for the case of finite depth, were given, substantially, by Airy, "Tides and Waves," Arts. 160. . . (1846). t Green, l.c. ante p. 366. t "Tides and Waves," Arts. 169, 170. 229] Numerical RemUs 357 results shew that a variation in the depth will have no appreciable influence, provided the depth everywhere exceeds (say) half the wave-length. 1 aechibA tanhibA cligk-^)^ c/to«* 000 1000 0000 0000 1000 •01 •998 •063 •260 •999 •02 •992 •126 •364 •997 •03 •983 •186 •432 •994 •04 •969 •246 •496 •990 •06 •963 •304 •662 •984 •06 •933 •360 •600 •977 •07 •911 •413 •643 •970 •08 •886 •464 •681 •961 •09 •869 •612 •716 •961 •10 •831 •667 •746 •941 •20 . •627 •860 •922 •823 •30 •297 •966 •977 •712 •40 •161 •987 •993 •627 •60 •086 •996 •998 •663 •60 •046 •999 •999 •616 •70 •026 1000 . 1000 •477 •80 •013 1000 1000 •446 •90 •007 1000 1-000 •421 100 •004 1000 1000 •399 00 •000 1000 1000 •000 Depth of Length of wave, in feet water, in feet 1 10 100 1000 10,000 Period of WA-VA. in RARonclfl 1 0442 1873 17-646 176-33 1763-3 10 0442 1-398 6-923 6680 667-62 100 0-442 1398 4-420 1873 176-46 1000 0442 1398 4-420 1398 6923 10,000 0442 1398 4-420 1398 44-20 1 Depth of • Length of wave, in feet » 1 water, in feet 1 10 100 1000 10,000 00 Wave- velocity, L 6-339 i 6667 in feet per sARond 1 2-262 6-671 6-671 6-671 10 2-262 7-154 16-88 17-92 17-93 17-93 100 2-262 7164 22-62 63-39 66-67 66-71 1000 2-262 7-164 22-62 71-64 168-8 179-3 10,000 2-262 7-164 22-62 7164 226-2 6671 358 Surface Waves [chap, ix We remark, finally, that the theoiy of progressiTe waves may be obtained, without the intermediary of standing waves, by assnming at once, in place of Art. 228 (1), if> = Pc'<'*-»*>. .'. (10) « The conditions to be satisfied by P are exactly the same as before, and we easily find, in real form, ly = a sin (fee — at), (11) . aacoshA;(y + A) ,, ^. ,^^. ^°f coshit ^ C08(fa:-at). (12) with the same determination of a as before. From (12) all the preceding results as to the motion of the rudividual particles can be inferred without difficulty. 230. The energy of a system of standing waves of the simple-harmonic type is easily found. If we imagine two vertical planes to be drawn at unit distance apart, parallel to a^, the potential energy per wave-length of the fluid between these planes is J Substituting the value of i) from Art. 228 (7), we obtain IgfM'^X . sin« (o< + e) (1) The kinetic energy is, by the formula (1) of Art. 61, *'/: [*i]...*'- Substituting from Art. 228 (8), and remembering the relation between q and i, we obtain Igpa^X . cos* (oi + c) (2) The total energy, being the sum of (1) and (2), is constant, and equal to \gpa^\. We may express this by saying that the total energy per unit area of the water-surface is \gpa^. A similar calculation may be made for the case of progressive waves, or we may apply the more general method explained in Art. 174. In either way we find that the energy at any instant is half potential and half kinetic, and that the total amount, per unit area, is \gp(^- In other words, the energy of a progressive wave-system of amplitude a is equal to the work which would be required to raise a stratum of the fluid, of thickness a, through a height \a. 229-231] Energy 369 231. The theory of progressiye waves may be investigated, in a very compact manner, by the method of Art. 175*. Thus if <f>, if/ be the velocity- and stream-functions when the problem has been reduced to one of steady motion, we assume <f> + uff = - (a? + iy) + iae<*<*+<»J -i- iJ3e-<*<»+<>'>, whence c X - (ae-"*» - i8c*«') sin kx, r = ^ y 4- {ae-^v + pe^v) cos hx. (1) This represents a motion which is periodic in respect to x, superposed on a uniform current of velocity c. We shall suppose that ka and kp are small quantities; in other words, that the amplitude of the disturbance is small compared with the wave-length. The profile of the free surface must be a stream-line ; we will take it to be the line = 0. Its form is then given by (1), viz. to a first approximation we have y = (a H- j3) cos fee, (2) shewing that the origin is at the mean level of the surface. Again, at the bottom (y = — A) we must also have = const. ; this requires 06** -i- j3c-** = 0. The equations (1) may therefore be put in the forms " = — a; -I- C cosh k{y -\- h)m.nkx, - = — y + C' sinh k{y -\- h) cos kx. c The formula for the pressure is — -^-»©*-(i)] (3) P P A = const. — fly — -o {1 — 2A;C cosh k{y + h) cos kx}, if we neglect k^CK Since the equation to the stream-line = is y = C sinh kh cos kx, approximately, we have, along this line, - = const. 4- {kc^ coth kh — g) y. P (*) * Rayleigh, l.c, ante p. 252. 360 Surface Waves [chap, ix The condition for a free surface is therefore satisfied, provided 4 , tanhM ,^. "=^*— HfcT (^) This determines the wave-length (27r/A;) of possible stationary undulations on a stream of given uniform depth A, and velocity c. It is easily seen that the value of kh is real or imaginary according as c is less or greater than {ghy. If we impress on everything the velocity — c parallel to x, we get progressive waves on still water, and (5) is then the formula for the wave- velocity, as in Art. 229. When the ratio of the depth to the wave-length is sufficiently great, the formulae (1) become ^ = - « + j3e*y sin fee, ?^ = - y + jSe*" cos fee, (6) leading to 2 = const, -gy-%{l- 2kpe^y cos kx + A;«jS*e"«'} (7) If we neglect i*jS*, the latter equation may be written 2 = const. + (Ac* - j) y + hop (8) Henceif ^'==1' (^) the pressure will be uniform not only at the upper surface, but along every stream-line »= const.* This point is of some importance ; for it shews that the solution expressed by (6) and (9) can be extended to the case of any number of liquids of different densities, arranged one over the other in horizontal strata, provided the uppermost surface be free, and the total depth infinite. And, since there is no limitation to the thinness of the strata, we may even include the case of a heterogeneous liquid whose density varies continuously with the depth. Cf. Art. 235. 232. The method of the preceding Art. can be readily adapted to a number of other problems. 1*. For example, to find the velocity of propagation of waves over the common horizontal boundary of two masses of fluid which are otherwise unlimited, we may assume 5^ = - y + jSe*>' cos fee, ^ = - y + jSe-*>' cos fee, (1) where the accent relates to the upper fluid. For these satisfy the condition of irrotational motion, V*^ = ; and they give a uniform velocity c at a great * This conclusion, it must be noted, is limited to the case of infinite depth. It was first remarked by Poisson, Lc. post p. 373. 231-232] Oscillations of Superposed Liquids 361 distance above and below the common surface, at which we have ^ = 0', = 0, say, and therefore y = jS cos fee, approximately. The pressure-equations are - = const. —gy—Kil — 2kpe^^ cos fee), ^. ,, y ...,....(2) ^ = const, -gy—^il-^- 2A^-*v cos fcc), which give, at the common surface, ^ = const. -(flr-*c«)y,] ^, r (3) S = const. - (flr 4- ic«) y, the usual approximations being made. The condition p = p' thus leads to c'-l'-^,> w a result first obtained by Stokes. The presence of the upper fluid has therefore the effect of diminishing the velocity of propagation of waves of any given length in the ratio {(1 — 8) /{I + «)}*, where 8 is the ratio of the density of the upper to that of the lower fluid. This diminution has a two-fold cause ; the potential energy of a given deformation of the common surface is diminished in the ratio 1 — 5, whilst the inertia is increased in the ratio 1 -t- «*. As a numerical example, in the case of water over mercury (s"^ = 13*6) the wave- velocity is diminished in the ratio '929. It is to be noticed, in this and in other problems of the kind, that there is a discontinuity of motion at the common surface. The normal velocity {dif//dx) is of course continuous, but the tangential velocity (— 9^/8y) changes from c (1 — i^ cos kx) to c (1 + ijS cos kx) as we cross the surface; in other words we have (Art. 151) a vortex-sheet of strength — 2kcp cos kx. This is an extreme illustration of the remark, made in Art. 17, that the free oscil- lations of a liquid of variable density are not necessarily irrotational. * Thia explains why the natural periods of oscillation of the common surface of two liquids of very nearly equal density are yery long compared with those of a free surface of similar extent. The fact was noticed by Benjamin Franklin in the case of oil oyer water; see a letter dated 1762 {CompUU Works, London, n. d., t. ii. p. 142). Again, near the mouths of some of the Norwegian fiords there is a layer of fresh over salt water. Owing to the comparatively small potential energy involved in a given deformation of the common boundary, waves of considerable height in this boundary are easily produced. To this cause is ascribed the abnormal resistance occasionally experienced by ships in those waters. See Ekman, "On Dead- Water/* Scientific ResuUs of the Norwegian North Polar Expedition, pt. xv. Christiania, 1904. 362 Surface Waves [chap, ix If p < p\ the value of c is imaginary. The undisturbed equilibrium- arrangement is then unstable. 2°. The case where the two fluids are confined between rigid horizontal planes y = — A, y = A' is almost equally simple. We have, in place of (1), Jf , ^ sinh k{y-\-h) , 0' ^sinh k{y- h') , (5) leadiBgto e* = |.___^_^-£^^^_ (6) When kh and kh' are both very great, this reduces to the form (4). When Jch' is large, and kh small, we have c^=.(l^L)gh, .(7) the main effect of the presence of the upper fluid being now the change in the potential energy of a given deformation. 3°. When the upper surface of the upper fluid is free, we may assume t-^+z'-A^ir *'-'». ^ (,, — = — y + (i3 cosh hy + y sinh ky) cos kx, and the conditions to be satisfied at the common boundary, and at the free surface, then lead to the equation c* (/> coth kh coth kh' + p) - c^p (coth kh' + coth kh) | +(/>-" P') Is = 0. (9) Since this is a quadratic in c*, there are two possible systems of waves of any given length (27r/A). This is as we should expect, for when the wave- length is prescribed the system has virtually two degrees of freedom, so that there are two independent modes of oscillation about the state of equilibrium. For example, in the extreme case where p'/p is small, one mode consists mainly in an oscillation of the upper fluid which is almost the same as if the lower fluid were solidified, whilst the other mode may be described as an oscillation of the lower fluid which is almost the same as if its upper surface were free. The ratio of the amplitude at the upper to that at the lower surface is found to be ^ . (10) kc^ cosh kh' — g sinh kh' 232-233] Oscillations of Superposed Liquids 363 Of the yarious special cases that may be considered, the most interesting is that in which kh is large ; i.e. the depth of the lower fluid is great compared with the wave-length. Putting coth kh = 1, we see that one root of (9) is now c» = |, (11) exactly as in the case of a single fluid of infinite depth, and that the ratio of the amplitudes is e**'. This is merely a particular case of the general result stated at the end of Art. 231 ; it will in fact be found on examination that there is now no slipping at the common boundary of the two fluids. The second root of (9) is, on the same supposition, c« _ P-P i^ (12) and for this the ratio (10) assumes the value -(^-l)6-»*' (13) If in (12) and (13) we put M' ^^ oo , we fall back on a former case. If on the other hand we make hh' small, we find c« = (l - ^') <?A', (14) aiid the ratio of the ampUtndes is -te-)- <p / These problems were first investigated by Stokes*. The case of any number of superposed strata of different densities has been treated by Webbf and Oreenhill;]:. 233. As a further example of the method of Art. 231 let us suppose that two fluids of densities />, />', one beneath the other, are moving parallel to x with velocities 17, V\ respectively, the common surface (when undisturbed) being of course plane and horizontal. This is virtually a problem of small oscillations about a state of steady motion. The fluids being supposed unlimited vertically, we assume, for the lower fluid ^=- V{y-^^eo%kcl (1) and for the upper fluid f « - 17 ' {y - pe-^y cos fee}, (2) • "On the Theory of OsciUatory Waves," Camb. Trans, t. viii. (1847) [Papers, t. i. p. 212]. t Math. Tripos Papers, 1884. t "Wave Motion in Hydrodynamics/' Amer, Jawm, of Maih. i. iz. (1887). 364 Surface Waves [chap. IX the origin being at the mean level of the common surface, which is assumed to be stationary-, and to have the form y = j8 cos he (3) The pressure-equations give i-^'^-^-i^'i^-^'-'-^. \ P _ > = const. -ffy-iU'^(l-\- 2A^e-"»«' cos Jb), (*) whence, at the common surface, ?- const. 4- (kU^ — g) y. 2^ = const. - (kU'^ + g) y. P Since we must have p = p' over this surface, we get (5) (6) This is the condition for stationary waves on the common surface of the two currents Uy U\ It may be written / pU + p'U 'y^g p'-p' pp \ p-\-p' J k'p-\-p' (7) The quantity pU-hp'U' p-h P may be called the mean velocity of the two currents; and it appears that relatively to this the waves have velocities ± c, given by ..Au-ur, (8) where Cq denotes the wave-velocity in the absence of currents (Art. 232). If the relative velocity \U — U' \ oi the currents exceed a certain limit, given by (Cr-Z7')^ = f.^-~f" (9) k pp the value of c is imaginary, indicating instability. This upper limit diminishes indefinitely with the wave-length. This result would indicate that, if there were no modifying circumstances, the slightest breath of wind would be sufficient to ruffle the surface of water. We shall give, later, a more complete investigation of the present problem, taking account of capillary forces, which act in the direction of stability. 233-234] Waves on a Surface of Discontinuity 366 It appears from (9) that if /> = />', or if j = 0, the plane form of the surface is mistable for all wave-lengths. This result illustrates the state- ment, as to the instability of surfaces of discontinuity in a liquid, made in Art. 79*. When the ourrents are confined by fixed horizontal planes y= -h,y=h\we assume ^=-u[y-p Binhk "^*^}' ^'=-^'{y^P sinhk- ""'M- (10) The condition for stationary waves on the common surface is then found to be pU^ coth kh +p'U'^ ooth kW =| (p -p') (ll)t It appears on examination that the undisturbed motion is stable or unstable, according as ^_^.^gpcothfc;i.fp-cothfcft- ^^ ^j2j (pp' coth ifcA coth W)* where Cq is the wave- velocity in the absence of currents. When h and W both exceed half the wave-length, this gives practically the former criterion (9). 234. These questions of stability are so important that it is worth while to give the more direct method of treatment J. If <f> be the velocity-potential of a slightly disturbed stream flowing with the general velocity V parallel to x, we may write <f, = -Vx + <f,, , (1) where <fti is small. The pressure-formula is, to the first order, M'-"-^''l'+-- <^) and the condition to be satisfied at a bounding surface ^ = ^, where t] is small, is Jt'^^di^"^ ^^^ To apply this to the problem stated at the beginning of Art. 233, we assume, for the lower and upper fluids, respectively, ^j = Ce*J'+< (*«-'«, <^/ = (7V*y+»<**"'*> ; (4) with, as the equation of the common surface, -q = ae«*»-'« (5) * The instability was first remarked by Hehnholtz, Lc. ante p. 21. t Greenhill, /.e. ante p. 363. t Sir W. Tbomaon, " Hydrokinetic Solutions and Observations," Phil Mag, (4), t. zli. (1871) [Baltimore Lectures, p. 590]; Rayleigh, "On the Instability of Jets," Proc. Land. Math, 8oc, t. X. p. 4 (1878) [Papers, t. i. p. 361]. 366 Stirface Waves [chap, ix The continuity of the pressure at this surface requires, by (2), p{i {a -kU)C-j- ga) = p' {% {a -kU')C'-¥ga}\ (6) whilst the surface-condition (3) gives i(a-kU)a^kC, {{g - kU')a= - kC (7) Eliminating a, C, C\ we get p(a -kUf -V p' (a -kUy^ gk{p ^ p'), (8) whence f = pE±P^^ /{| . PjU^ ^ ^P_ (^7 - V'A. . . . .(9) k p-\-p \{kp^p. (p-¥pY^ \r leading to the same conclusions as in Art. 233. If (U--Uy> ^^'^f^\ \ (10) PP where Cq is the wave- velocity in the absence of currents, a is imaginary, of the form a ± ij3. The complete solution then contains a term with eP* as a factor, indicating indefinite increase of amplitude. If p = />', it is evident from (8) that a will be imaginary for all values of k. Putting U' = — U, we get a = ±ikU, (11) Hence, taking the real part of (5), we find -q = ae*^u* coBkx (12) The upper sign gives a system of standing waves whose height continually increases with the time, the rate of increase being greater, the shorter the wave-length. The case of p = p\ with U = U', is of some interest, as illustrating the flapping of sails and flags*. We may conveniently simplify the question by putting U = U' = 0; any common velocity may be superposed afterwards if desired. On these suppositions, the equation (8) reduces to a* = 0. On account of the double root the solution has to be completed by the method explained in books on Differential Equations. In this way we obtain the two independent solutions -q = 06***, <^i = 0, <^/ = 0, (13) and 7) = ate**^ <^i = - ? e*'' . e**^ <f>i = | e'^y . e**«. . . (14) The former solution represents a state of equilibrium; the latter gives a system of stationary waves with amplitude increasing proportionally to the time. In this form of the problem there is no physical surface of separation to begin with ; but if a slight discontinuity of motion be artificially produced, e.g. by impulses applied to a thin membrane which is afterwards dissolved, * Rayleigh, l.c. 234-235] Osculations of a HeterogeneoVfS Liquid 367 the discontinuity will peisist, and, as we have seen, the height of the corrugations will continually increase. The above method, when applied to the ease where the fluids are oonfined between two rigid horizontal planes y= -Ky^h', leads to p (<r -itJ7)« coth kh +p' {<r -kUy oath kh'=gk (p -/)» (IS) which is equivalent to Art. 233 (11). 235. The theory of waves in a heterogeneous liquid may be noticed, for the sake of comparison with the case of homogeneity. The equilibrium value p^ of the density will be a function of the vertical co-ordinate (y) only. Hence, writing P = Po + p\ /> = />o + />', (1) where p^ is the equilibrium pressure, the equations of motion, viz. du dp ,dv dp ,^, i+«i+'i-»- • w , du dp' dv dp' , ... become p._ = _^. p„_«_^_^p (4) |' + «| = 0' (5) small quantities of the second order being omitted. The fluid being incom- pressible, the equation of continuity retains the form du dv ai + ^ = «' (6) so that we may write " dy' "-dx (^) Eliminating p' and p' we find* '** - ^/^ if - 4-1} = » At a free surface we must have Dp/Dt = 0, or (8) ¥-«|'-^'.| <9> Hence, and from (4), we must have ^-»g OT at such a surface. * Cf. Love, "Wave Motion in a Heterogeneous Heavy Liquid,'* Proc. Land, Math, Soc, t. xxii. p. 307 (1891). 368 Surf (ice Waves [chap, ix To investigate cases of wave-motion we assume that X g<«r«-*«) (11) The equation (8) becomes whilst the surface-condition takes the form |-$V-« (>») These are satisfied, whatever the vertical distribution of density, by the assumption that varies as e^^^ provided <^ = i7* (U) For a fluid of infinite depth the relation between wave-length and period is then the same as in the case of homogeneity (cf. Art. 231), and the motion is irrotational. For farther investigations it is necessary to make some assumption as to the relation between po and y. The simplest is that Po«c-^^ (16) in which case (12) takes the form p-'^-Kt-f)+-»- ('•) The solution is V^=(^6^'«'+J5e^»') «<<'*-*«>, (17) where Xj, X, are the roots of X«-pX + (^-l)jk«=0 (18) We may apply this to the oscillations of liquid filling a closed rectangular vessel*. The quantity h may be any multiple of n-/^, where / denotes the length. If the equations to the horizontal boundaries be y =0, y = A, the condition 8^/9a; =0 gives ^+J5=0, i4c^»*+J5«^=0 (19) whence c<^i-^''* = l, or X^ -X,=2t>Tr/A, (20) where * is integral. Hence, from (18), Xi =i/3+wW^, X2=i/3 -iB7rl\ (21) and therefore (g-l)i«=X,X.=i/3«+'^* (22) We verify that o- is real or imaginary, t.e. the equilibrium arrangement is stable or unstable, according as /3 is positive or negative, i.e. according as the density diminishes or increases upwardsf. * Rayleigh, "Investigation of the Character of the Equilibrium of an Incompressible Heavy Liquid of Variable Density," Vroc, Lond. McUh, 8oc. t. adv. p. 170 [Papers, t. ii. p. 200]. Reference may also be made to a paper "On Atmospheric Oscillations," Proc. Roy. 8oc. t. Ixzxiv. pp. 566, 671 (1910), where another law of density is considered. t The case of waves on a liquid of finite depth is discussed by Love (/.c). See also Bumside, *'0n the Small Wave-Motions of a Heterogeneous Fluid under Gravity," Proe. Lond, Math. Soc. t. xz. p. 392 (1889). 235-236] Theory of Wave-Groups 369 236. The investigations of Arts. 227-234 relate to a special type of waves ; the profile is simple-harmonic, and the train extends to infinity in both directions. But since all our equations are linear (so long as we confine ourselves to a first approximation), we can, with the help of Fourier's Theorem, build up by superposition a solution which shall represent the effect of arbitrary initial conditions. Since the subsequent motion is in general made up of systems of waves, of all possible lengths, travelling in either direction, each with the velocity proper to its own wave-length, the form of the free surface will continually alter. The only exception is when the wave-length of every system which is present in sensible amplitude is large compared with the depth of the fluid. The velocity of propagation, viz. y/{gh\ is then independent of the wave-length, so that in the case of waves travelling in one direction only, the wave-profile remains unchanged in form as it advances (Art. 170). The effect of a local disturbance of the surface, in the case of infinite depth, will be considered presently; but it is convenient to introduce in the first place the very important conception of * group- velocity,' which has application, not only to water-waves, but to every case of wave-motion where the velocity of propagation of a simple-harmonic train varies with the wave-length. It has often been noticed that when an isolated group of waves, of sensibly the same length, is advancing over relatively deep water, the velocity of the group as a whole is less than that of the individual waves composing it. If attention be fixed on a particular wave, it is seen to advance through the group, gradually dying out as it approaches the front, whilst its former place in the group is occupied in succession by other waves which have come forward from the rear *. The simplest analytical representation of such a group is obtained by the superposition of two systems of waves of the same amplitude, and of nearly but not quite the same wave-length. The corresponding equation of the free surface will be of the form 7y = a sin (Jcx — <rf) + a sin (k>x — at) = 2a cos{i (h-K)x-\(G- a) t) sin {J (k-^W)x-\(a + a')t}. (1) If Jk, h' be very nearly equal, the cosine in this expression varies very slowly with x\ so that the wave-profile at any instant has the form of a curve of sines in which the amplitude alternates gradually between the values and 2a. The surface therefore presents the appearance of a series of groups of * Scott Russell, "Report on Waves," Brit. Aaa. Rep, 1844, p. 369. There is an interesting letter on this point from W. Fronde, printed in Stokes* Seientiflc Correspondence, Cambridge, 1907, t. ii p. 166. L. H. 24 370 Surface Waves [chap, ix waves, separated at equal intervals by bands of nearly smooth water. The motion of each group is then sensibly independent of the presence of the others. Since the distance between the centres of two successive groups is 27r/(A; — i'), and the time occupied by the system in shifting through this space is 27r/((7 — a\ the group- velocity (i7, say) is = (a — ct')/(* — *')» o^" "-S (^> ultimately. In terms of the wave-length A (= 27r/A;), we have where c is the wave- velocity. This result holds for any case of waves travelling through a uniform medium. In the present application we have i = (|tanhMr, (4) and therefore, for the group- velocity, ~5F^*K^^Snh2A:A) ^^^ The ratio which this bears to the wave- velocity c increases as AA diminishes, being \ when the depth is very great, and unity when it is very small, compared with the wave-length. The above explanation seems to have been first given by Stokes*. The extension to a more general type of group was made by Rayleighf and Gouyf . The argument of these writers admits of being put very concisely. Assuming a disturbance y = SC cos (a< - *a? 4- €), (6) where the summation (which may of course be replaced by an integration) embraces a series of terms in which the values of a, and therefore also of A;, vary very slightly, we remark that the phase of the typical term at time i-\- ^i and place x-\- Lx differs from the phase at time t and place x by the amount uNi — kAx, Hence if the variations of a and k from term to term be denoted by 8a and 8k, the change of phase will be sensibly the same for all the terms, provided &7 A< - SA; Ax = (7) * Smith's Piize Examination, 1876 [Papers, t. v. p. 362] . See also Rayleigh, Theory of Sound, Art. 191. t Nature, t. xxv. p. 62 (1881) [Papers, t. i. p. 640]. X "Sur la Vitesse de la lumi^re/' Ann. de Chim, et de Phys. t. xvi. p. 262 (1889). It has recently been pointed out that the theory had been to some extent anticipated by Hamilton, working from the optical point of view, in 1839; see Havelock, Cambridge Tracts, No. 17 (1914), p. 6. 236] Group- Vdocitp 371 The group as a whole therefore travels with the velocity Ax da (8) Another derivation of (3) can be given which is, perhaps, more intuitive. In a medium such as we are considering, where the wave- velocity varies with the frequency, a limited initial disturbance gives rise in general to a wave- system in which the different wave-lengths, travelling with different velocities, are gradually sorted out (Arts. 238, 239). If we regard the wave-length X as a function of x and <, we have (9) since A does not vary in the neighbourhood of a geometrical point travelling with velocity U ; this is, in fact, the definition of U. Again, if we imagine another geometrical point to travel with the waves, we have 3A 3A _ V 3c _ V (fc 3A dt dx dx dXdx* (10) the second member expressing the rate at which two consecutive wave-crests are separating from one another. Combining (9) and (10), we are led, again, to the formula (3) *. This formula admits of a simple geometrical representation f. If a curve be constructed with A as abscissa and c as ordinate, the group- velocity will be represented by the intercept made by the tangent on the ♦ See a paper "On Group- Velocity," Proc. Lond, Math. 8oc, (2), t. i. p. 473 (1904). The subject is further discussed by G. Green, "On Group- Velocity, and on the Propagation of Waves in a Dispersive Medium," Proc. R. 8. Edin. t. ixix. p. 445 (1909). t Manch. Mem. t. xliv. No. 6 (1900). 24—2 372 SurfcLce Waves [chap, ix axis of c. Thus, in the figure, PN represents the wave- velocity for the wave-length ON, and OT represents the group- velocity. The frequency of vibration, it may be noticed, is represented by the tangent of the angle PON, In the case of gravity- waves on deep water, c « A* ; the curve has the form of the parabola y* = 4ax, and OT = \PN, i.e., the group- velocity is one- half the wave- velocity. *237. The group- velocity has moreover a dynamical, as well as a geometrical, significance. This was first shewn by Osborne Reynolds*, in the ca<se of deep-water waves, by a calculation of the energy propagated across a vertical plane. In the case of infinite depth, the velocity-potential corresponding to a simple-harmonic train f] = a mih (x — d) (11) is <f> = ace^^ cos k{x — ct), » (12) as may be verified by the consideration that for y = we must have 37^/3^ = — d<f>/dy. The variable part of the pressure is pd<f>/dt, if we neglect terms of the second order, so that the rate at which work is being done on the fluid to the right of the plane x is - [ P^<iy= pa^k^c^ 8in2 k(x-ct)[ e^^y dy = \gpa^c sin* k(x — ct), (13) since c* = gjk. The mean value of this expression is \gpa^c. It appears on reference to Art. 230 that this is exactly one-half of the energy of the waves which cross the plane in question per unit time. Hence in the case of an isolated group the supply of energy is sufficient only if the group advance with AaZ/the velocity of the individual waves. It is readily proved in the same manner that in the case of a finite depth h the average energy transmitted per unit time isf i^''"''' (i + sJk) (1^) which is, by (5), the same as \9pa'^^ (15) Hence the rate of transmission of energy is equal to the group- velocity, d (kc)/dk, found independently by the former line of argument. * "On the Rate of Progression of Groups of Waves, and the Rate at which Energy is Transmitted by Waves," Nahire, t. xvi. p. 343 (1877) [Papers, t. i. p. 198]. Reynolds also constructed a model which exhibits in a very striking manner the distinction between wave- velocity and group- velocity in the case of the transverse oscillations of a row of equal pendulums whose bobs are connected by a string. t Rayleigh, "On Progressive Waves," Proc, Lond, Math. Soc. t. ix. p. 21 (1877) [Papers, t. L p. 322]; Theory of Sound, t. i. Appendix. 23ft-238] Propagation of Energy 373 This identification of the kinematical group- velocity of the preceding Art. with the rate of transmission of energy may be extended to all kinds of waves. It follows indeed from the theory of interference groups (p. 369), which is of a general character. For let P be the centre of one of these groups, Q that of the quiescent region next in advance of P. In a time t which extends over a number of periods, but is short compared with the time of transit of a group, the centre will have moved to P\ such that PP' = [7t, and the space between P and Q will have gained energy to a corresponding amount. Another investigation, not involving the notion of * interference,' was given by Rayleigh (i.e.). Prom a physical point of view the group-velocity is perhaps even more important and significant than the wave- velocity. The latter may be greater or less than the former, and it is even possible to imagine mechanical media in which it would have the opposite direction ; i.e. a disturbance might be propagated outwards from a centre in the form of a group, whilst the in- dividual waves composing the group were themselves travelUng backwards, coming into existence at the front, and dying out as they approach the rear*. Moreover, it may be urged that even in the more familiar pheno- mena of Acoustics and Optics the wave-velocity is of importance chiefly so far as it coincides with the group-velocity. 238. The theory of the waves produced in deep water by a local disturbance of the surface was investigated in two classical memoirs by Cauchyf and Poisson;!:. The problem was long regarded as difiBicult, and even obscure, but in its two-dimensional form, at all events, it can be pre- sented in a comparatively simple aspect. It appears from Arts. 40, 41 that the initial state of the fluid is deter- minate when we know the form of the boundary, and the boundary-values of the normal velocity d<f>/dn, or of the velocity-potential <^. Hence two forms of the problem naturally present themselves; we may start with an initial elevation of the free surface, without initial velocity, or we may start with the surface undisturbed (and therefore horizontal) and an initial distribution of surface-impulse {fxf>o). If the origin be in the undisturbed surface, and the axis of y be drawn vertically upwards, the typical solution for the case of initial rest is T) = cos at cos kXy (1) (f) = g 6*1' cos kx, (2) d provided a^ =, gk, (3) • Proc. Land. Math. 8oc. (2), t. i. p. 473. t l.c. ante p. 16. } "M^moire sur )a thdorie dee ondes," Mim, de VAcad. Roy. des Sciences, t. i. (1816). 374 Surface Waves [chap, ix in accordance with the ordinary theory of 'standing' waves of simple- harmonic profile (Art. 228). If we generalize this by Fourier's double-integral theorem /(a;) = - I dk\ /(a) cos k{x — a)da, (4) then, corresponding to the initial conditions ri^f(x), <^o = 0, (5) where the zero suffix indicates surface- value (y = 0), we have rj = - I cos atdkl f (a) cos k {x — a) da, (6) TT j J -00 £ f" smat^^^^^ r j.^^^ coBk{x-a)da (7) rrj a a J -a> I If the initial elevation be confined to the immediate neighbourhood of the origin, so that/ (a) vanishes for all but infinitesimal values of a, we have, assuming '" f{a)da=h (8) / -00 This may be expanded in the form ^ ^ r Sinore ^f:y^^^j^^j^ (9) ^ = ^r\l ~^* Jfc + ^A:*- . . .1 e^^ cos kxdk, . . . .(10) where use is made of (3). If we write — y = r cos &, a: = r sin 0, (11) we have, y being negative*, / n I c^^ cos fcx/c"afc = - c*v cos kxk^'dk = -^ cos (w + 1) (9, (12) so that (10) becomes , at (cob d l.-_^«, cos2& . 1 ,- --, cos3d ) ,,^. a result which is easily verified. From this the value of t) is obtained by Art. 227 (5), putting d = ± lir. Thus, for x > 0, ^ iTa;|2x 3.5\2x) "^ 3 . 5 . 7 . 9 V2a;/ •••]• "^ '' * This formula may be dispensed with. It is sufficient to calculate the value of 4> at points on the vertical axis of symmetry ; its value at other points can then be written down at once by a property of harmonic functions (cf. Thomson and Tait, Art. 498). I That the effect of a concentrated initial elevation of sectional area Q would be of the form might have been foreseen from consideration of * dimensions.* 238] Cauchy-Poisson Wave-Problem 375 It is evident at once that any particular phase of the surface disturbance, e.g., a zero or a maximum or a minimum of t^, is associated with a definite value of \gt^jx, and therefore that the phase in question travels over the surface with a constant acceleration. The meaning of this somewhat remarkable result will appear presently (Art. 240). The series in (14) is virtually identical with one (usually designated by -M *) which occurs in the theory of FresnePs diffraction-integrals. In its present form it is convenient only when we are dealing with the initial stages of the disturbance; it converges very slowly when \gt^jx is no longer small. An alternative form may, however, be obtained as follows. The surface- value of <f> is, by (9), <f>o= I cos kxdk = -jl sin f Vcft\Ao—\ sin ( (rfj dorj- (15) Putting ^ = ;!h©' (^'^ wefind l^'sin (— + erf) dcr = ?- r sin (S« - co«) d$, (17) J*sin (?-?« erf) d(7= ^ r sin (i« - co2) (iS, (18) where "^ = (feT ^^^^ Hence <^o = ~ ^f "sin (5« - co^) (?{ (20) From this the value of 17 is derived by Art. 227 (5) ; thus V = ^ cos ({2 _ co«) dS TTX^J = ^ jcosco« f "cosC«(i$ + sinco2 Tsin C^d^l (21) Trrc* ( ^ / } This agrees with a result given by Poisson. The definite integrals are practically of Fresnel's forms f, and may be considered as known functions. ♦ Cf. Rayleigh, Papers, t. iii. p. 129. I In terms of a usual aotation we have where C (u) = I cob i TU^du, S (ti) = / sin J tu' du, the upper limit of integration being u=J(2jt).w. Tables of C(«) and S(ii), computed by Gilbert and others, are giyen in most books on Physical Optics. 376 Surface Waves [OHAP. IX Lommel, in his researches on Diffraction*, has given a table of the function 1- .2 + «)«D O«0»i«t7 (22) which is involved in (14), for values of z ranging from to 60. We are thus enabled to delineate the first nine or ten waves with great ease. The figure below shews the variation of i] with the time, at a particular place; for different places the intervals between assigned phases vary as ^/x, whilst the corresponding elevations vary inversely as x. The diagrams on p. 377, on the other hand, shew the wave-profile at a particular instant ; at different times, the horizontal distances between corresponding points vary as the square of the time that has elapsed since the beginning of the disturbance, whilst corresponding elevations vary inversely as the square of this time. [The UDit of the horizontal scale is ^{2x/g). That of the vertical scale is Q/nx, if Q be the sectional area of the initially elevated fluid.] When gt^l^x is large, we have recourse to the formula (21), which makes approximately, as found by Poisson and Cauchy. This is in virtue of the known formulae Jo Jo 2V2' (24) Expressions for the remainder are also given by these writers. Thus Poisson obtains, substantially, the semi-convergent expansion V = 9^t 2*^* -♦HS -.^{^«-^-^-K^«) ' /2x + 1.3.5.7.9 (=r \gt 2 i * * * f (25) * **Bie Bengungserscheinungen geradlinig begrenzter Sohirme," Ahh, d. k. Bayer, Akad, d, Wiss, 2^ CI. t. XV. (1886). 238] Waves dtce to a Local JSlevation 377 -50-. -100' 0-d 0.4 OS 7 600i 400 • 30a 20a 100- • -10a -200" -300. -400-^ [The unit of the horizontal scales is igt*. That of the vertical scales is 2Q/irgi^.] 378 Surface Waves [chap, ix This is deriyed as follows. We have \^y^^'^^ dC= fl e' ^^'-'^ di - J]] e' (^*--"> dC = iV^e - ' +_+^-^^^ + ^_^ + ..., (26) by a series of partial integrations. Taking the real part, and substituting in the first lino of (21), we obtain the formula (25). 239. In the case of initial impulses applied to the surface, supposed horizontal, the typical solution is f}^ = cos d e^^ cos kx, (27) ri = sin <rf cos kx, (28) 9P with a* = gk as before. Hence, if the initial conditions be f4^^F{xl t; = 0, (29) 1 /•« /■« we have <i = — cos a< c*" dk\ F (a) cosk (x — a) da, (30) ^P.' J -00 Ti =s I aBmaldk] F (a) cos i (a? — a) ia (31) '"'9PJ ^ -00 For a concentrated impulse acting at the point » = of the surface, we have, putting r F(a)da=l, (32) J —00 1 f* xk =z — I COS att 6*" cos kxdk (33) ^/>i This integral may be treated in the same manner as (9) ; but it is evident that the results may be obtained immediately by performing the operation l/gp . d/dt upon those of Art. 238. Thus from (13) and (14) we derive . 1 (cos^ , -Cos2& 1 ,1 ,«v.cos3& I ,,.. ^^Trpx^ll 1.3.5V2X/ ■^1.3.5.7.9V2x/ •••!• •• (35) * * With the help of the theory of ' dimenaions * it is easily seen d priori that the effect of a concentrated initial impulse P (per unit breadth) is necessarily of the form Pt 23&-239] Waves due to a Local Impulse 379 The series in (35) is related to the function + (36) 1.3 1.3.5.7 ' 1.3.5.7.9.11 which has also been tabulated by Lommel. If we denote the series (22) and (36) by Si and S,, respectively, we find 32« 52* ^ 1.3.5*^1.3.5.7.9 -,..=J(1 + Si-2jS2), ....(37) so that the forms of the first few waves can be traced without difficulty. The annexed figure shews the rise and fall of the surface at a particular place; for different places the time-iptervals between assigned phases vary as \^x, as in the former case, but the corresponding elevations now vary [The unit of the horizontal scale is »J{2x/g). That of the yertical scale is P /2 — . / — , where P represents the total initial impulse.] npz \ gz inversely as x^. In the diagrams on p. 380, which give an instantaneous view of the wave-profile, the horizontal distances between corresponding points vary as the square of the time, whilst corresponding ordinates vary inversely as the cube of the time. For large values of \gt^lx, we find, performing the operation Ijgp . djdt upon (23), '-.fcHS-"-© <'«> ap{]froximately. Surface Waves [oaiP. IX 7 1O00O [The unit of the horizontal scalee is liTt*. That of the vertical scales is - — r-^. The upper curve, if continued to the right, would cross the axis of x and would thereafter be indiBtingui^hsble from it on the present scale.} 23»-240] Interpretation of Results 381 240. It remains to examine the meaning and the consequences of the results above obtained. It wiU be sufficient to consider, chiefly, the case of Art. 238, where an initial elevaiion is supposed to be concentrated on a line of the surface. At any subsequent time t the surface is occupied by a wave-system whose advanced portions are delineated on p. 377. For sufficiently small values of X the form of the waves is given by (23) ; henoe as we approach the origin the waves are found to diminish continually in length, and to increase continually in height, in both respects without limit. As t increases, the wave-system is stretched out horizontally, proportionally to the square of the time, whilst the vertical ordinates are correspondingly diminished, in such a way that the area i ridx included between the wave profile, the axis of x, and the ordinates corre- sponding to any two assigned phases (i.e., two assigned values of co) is constant*. The latter statement may be verified immediately from the mere form of (14) or (21). The oscillations of level, on the other hand, at any particular place, are represented on p. 376. These follow one another more and more rapidly, with ever increasing amplitude. For sufficiently great values of t, the course of these oscillations is given by (23). In the region where this formula holds, at any assigned epoch, the changes in length and height from wave to wave are very gradual, so that a considerable number of consecutive waves may be represented approxi- mately by a curve of sines. The circumstances are, in fact, all approximately reproduced when Ag = 2«r (39) Hence, if we vary t alone, we have, putting At = t, the period of oscillation, ^=-^5 (*o) whilst, if we vary x alone, putting Ax = — A, where A is the wave-length, we find A = ^ (41) * This stfttement does not apply to the case of an initial impuUe, The corresponding pro- position then is that taken between assigned values of w, is constant. This appears from (34). 382 Surface Waves [chap, ix The wave- velocity is to be found from Ag = 0; ., (42) thiflgives ^ = T = \/S (*^) by (41), as in the case of an infinitely long train of simple-harmonic waves of length A. We can now see something of a reason why each wave should be con- tinually accelerated. The waves in front are longer than those behind, and are accordingly moving faster. The consequence is that aU the waves are continually being drawn out in length, so that their velocities of propagation continually increase as they advance. But the higher the rank of a wave in the sequence, the smaller is its acceleration. So far, we have been considering the progress of individual waves. But, if we fix our attention on a group of waves, characterized as having (approxi- mately) a given wave-length A, the position of this group is regulated according to (43) by the formula -.-iVt-- <") i.e., the group advances with a constant velocity equal to hcdf that of the component waves. The group does not, however, maintain a constant amplitude as it proceeds ; it is easily seen from (23) that for a given value of A the amplitude varies inversely as ^/x. It appears that the region in the immediate neighbourhood of the origin may be regarded as a kind of source, emitting on each side an endless succession of waves of continually increasing amplitude and frequency, whose subsequent careers are governed by the laws above explained. This persistent activity of the source is not paradoxical ; for our assumed initial accumulation of a finite volume of elevated fluid on an infinitely narrow base implies an unlimited store of energy. In any practical case, however, the initial elevation is distributed over a band of finite breadth ; we will denote this breadth by I. The disturbance at any point P is made up of parts due to the various elements, Sa, say, of the breadth I; these are to be calculated by the preceding formulae, and integrated over the- breadth of the band. In the result, the mathematical infinity and other perplexing peculiarities, which we meet with in the case of a concentrated line-source, disappear. It would be easy to write down the requisite formulae, but, as they are not very tractable, and contain nothing not implied in the preceding statement, they may be passed over. It is more instructive to examine, in a general way, how the previous results will be modified. 240] Interpretation of Results 383 The initial stages of the disturbance at a distance Xy such that Ijx is small, will evidently be much the same as on the former hypothesis; the parts due to the various elements 8a will simply reinforce one another, and the result will be sufficiently expressed by (14) or (23) provided we multiply by / 00 / (a) da. -00 i.e., by the sectional area of the initially elevated fluid. The formula (23), in particular, will hold when \gt^lx is large, so long as the wave-length A at the point considered is large compared with i, i.e., by (41), so long as \gt'^lx . Ijx is small. But when, as t increases, the length of the waves at x becomes comparable with or smaller than I, the contributions from the different parts of I are no longer sensibly in the same phase, and we have something analogous to 'interference' in the optical sense. The result will, of course, depend on the special character of the initial distribution of the values of /(a) over the space Z*, but it is plain that the increase of amplitude must at length be arrested, and that ultimately we shall have a gradual dying out of the disturbance. There is one feature generally characteristic of the later stages which must be more particularly adverted to, as it has been the cause of some perplexity ; viz. a fluctuation in the amplitude of the waves. This is readily accounted for on 'interference' principles. As a sufficient illustration, let us suppose that the initial elevation is uniform over the breadth I, and that we are considering a stage of the disturbance so late that the value of A in the neighbourhood of the point x under consideration has become small compared with Z. We shall evidently have a series of groups of waves separated by bands of comparatively smooth water, the centres of these bands occurring whenever I is an exact multiple of A, say I = nA. Substituting in (41), we find ^-i^J& <*^) I i.e., the bands in question move forward with a constant velocity, which is, in fact, the group-velocity corresponding to the average wave-length in the neighbourhood f . * Cf. Bumside, **0n Deep-water Waves resulting from a Limited Original Disturbance/* Proc. Land, Maih. 8oe, t. zx. p. 22 (1888). I This fluctuation was first pointed out by Poisson, in the particular case where the initial elevation (or rather depression) has a parabolic outline. The preceding investigations have an interest extending beyond the present subject, as shewing how widely the effects of a single initial impulse in a dispersive medium (t.e., one in which wave-velocity varies with wave-length) may differ from what takes place in the case of sound, or in the vibrations of an elastic solid. The above discussion is taken, with some modifica- tions, from a paper "On Deep- Water Waves," Proc, Lond. Math. Soe. (2), t. ii p. 371 (1904), where also the effect of a local periodic pressure is investigated. 384 Surface Waves [chap, ix The ideal solution of Art. 238 necessarily fails to give any information as to what takes place at the origin itself. To illustrate this point in a special case, we may assume /(«)=?647- (*«) the formula (7) then gives A=?^ f^^ «*(»-») 008 fa dft .(47) The surface-elevation at the origin is ri=^ r COB at e-»dk=^^ T cos crU"^*''' <rda=^ ~ T sin (rte^*^^'' da. . .(48) ' n Jo ^g Jo irgdtjo By a known formula we have* 1*6-^ Bin 2fizdx=e'-fi*r€?^dx (49) Hence, putting »' =gt*/ib, (50) we find ''=^<^-*"''*j ^^"^f^"^*"'*'/"^'^) (^^) Hence ^(^^"'^^ "S/J^^' ^^2) shewing that rjff^ steadily diminishes as t increases. Hence rj can only change sign once. The form of the integrals in (48) shews that 17 tends finally to the limit zero ; and it may be proved that the leading term in its asymptotic value is - 2QlirgtK One noteworthy feature in the above problems is that the disturbance is propagated inatantdneonsly to all distances from the origin, however great. Analytically, this might be accounted for by the fact that we have to deal with a synthesis of waves of all possible lengths, and that for infinite lengths the wave- velocity is infinite. It has been shewn, however, by Rayleighf that the instantaneous character is preserved even when the water is of finite depth, in which case there is an upper limit to the wave- velocity. The physical reason of the peculiarity is that the fluid is treated as incompressible, so that changes of pressure are propagated with infinite velocity (cf . Art. 20). When compressibility is taken into account a finite, though it may be very short, interval elapses before the disturbance manifests itself at any point];. 241. The space which has been devoted to the above investigation may be justified by its historical interest, and by the consideration that it deals with one of the few problems of the kind which can be solved completely. It was shewn, however, by Kelvin that an approximate representation of /; * This formula presents itself as a subsidiary result in the process of evaluating •00 e'''*coB2pxdx by a contour integration. t "On the Instantaneous Propagation of Disturbance in a Dispersive Medium, . . .," PhU, Mag. (6), t. xviii. p. 1 (1909) [Papers, t. v. p. 614]. See also Pidduck, "On the Propagation of a Disturbance in a Fluid under Gravity," Proc, Boy. Soc. A, t. Ixxxiii. p. 347 (1910). X Pidduck, "The Wave-Problem of Cauohy and Poisson for Finite Depth and slightly GozDpressible Fluid," Proc, Boy. Soc. A, t. Ixxxvi. p. 396 (1912). 240-241] Kelvin's Approxirnation 385 the more interesting features can be obtained by a simpler process, which is moreover of very general application*. The method depends on the approximate evaluation of integrals of the type e^f^^^dx (1) w = I <j>{x) J a It is assumed that the circular function goes through a large number of periods within the range of integration, whilst <f> (x) changes comparatively slowly ; more precisely it is assumed that, when / {x) changes by 27r, <f> (x) changes by only a small fraction of itself. Under these conditions the various elements of the integral will for the most part cancel by annulling interference, except in the neighbourhood of those values of x, if any, for which / {x) is stationary. If we write a = a + ^, where a is a value of x, within the range of integration, such that/' (a) = 0, we have, for small values of ^, /(«)=/(o) + if«Aa). (2) approximately. The important part of the integral, corresponding to values of a; in the neighbourhood of a, is therefore equal to <l> (a) 6<^'«» r e*</'(»)-«*d^, (3) J -00 approximately, since, on account of the fluctuation of the integrand, the extension of the limits to ± oo causes no appreciable error. Now by a known formula (Art. 238 (24)) we have r «*""'^'^^=^-^*=^-«**" w Hence (3) becomes yrt^?\ ■e«l/c)'^> (5) V I i/" (a) where the upper or lower sign is to be taken in the exponential according as /"(a) is positive or negative. If a coincides with one of the limits of integration in (1), the limits in (3) will be replaced by and oo , or — oo and 0, and the result (5) is to be halved. If the approximation in (2) were continued, the next term would be i^V"(a); til© foregoing method is therefore only vaUd under the condition that f/'" (a)//" (a) must be small even when ^y ' (a) is a moderate multiple of 277. This requires that the quotient should be small. * Sir W. Thomson, "On the Waves produced by a Single Impulse in Water of any Depth, or in a Dispersive Medium," Proc. B. S, t. xlii. p. 80 (1887) [PaperSy t. iv. p. 303]. The method of treating integrals of the type (1) had however been suggested by Stokes in his paper "On the Numerical Calculation of a Class of Definite Integrals and Infinite Seriee," Camb. Trans. t. ix. (1850) [Papers, t. ii. p. 341, footnote]. L. H. 25 386 Surface Waves [chap, ix In wave-problems of the kind now under consideration, the effect of a concentrated initial disturbance is given by formulae of the type ^ = 1 f *^ (*) e<(-i-*«» ijfc + ^ r^ (jfc) e'^-*+*»» dk (6) where a is a known function of k^ viz. 29r/<7 is the period of oedllatioii in a train of simple-harmonic waves of length ^fnjk. It is understood that in the end only the real part of the expression is to be retained. The two terms in (6) represent the results of superposing trains of simple- harmonic waves of all possible lengths, travelling in the positive and negative directions of a;, respectively. If, taking advantage of the symmetry, we confine our attention to the region lying to the right of the origin, the ex- ponential in the first integral will alone, as a rule*, admit of a stationary value or values, viz. when 'dk^' <^) This detenxiines it as a function of x and t, and we then find, in accordance with (5), where the ambiguous sign follows that of cPafdk^. The approximation postulates the smallness of the ratio dV/dP ^ V{M^V^**I*} (9) Since y (10) by (7), it appears that the wave-length and the period in the neighbourhood of the point x at time t are 27r/k and 27r/or, respectively. The relation (7) shews that the wave-length is such that the corresponding jrow^velocity (Art. 236) is x/t. The above process, and the result, may be illustrated by various graphical constructionsf. The simplest, in some respects, is based on a slight modification of the diagram of Art. 236. We construct a curve with X as abscissa and ct as ordinate, where t denotes the time that has elapsed since the beginning of the disturbance. To ascertain the nature of the wave- system in the neighbourhood of any point z, we measure off a length OQ, equal to z, along the axis of ordinates. If PN be the ordinate corresponding to any given abscissa X, the * If the group- velocity were negative, as in some of the artificial cases referred to in Art. 237, the second integral would be the important one. f Proc. of the nth IrUem. Congress of McUhematicians, Cambridge, 1912, p. 281. 241] Geometrical lUustrations 387 phaso of the disturbance at x, due to the elementary wave-train whose wave-length is X, will be given by the gradient of the Une QP ; for if we draw QB parallel to ON, we have ) PR _ PN-OQ _ct-x at-kx QB~ ON " X " 2ir (11) Hence the phase will be stationary if QP be a tangent to the curve ; and the predominant wave-lengths at the point x are accordingly given by the abscissae of the points of contact of the several tangents which can be drawn from Q, These are characterized by the property that the group-velocity has a given value x/t. If we imagine the point Q to travel along the straight line on which it lies, we get an indication of the distribution of wave-lengths at the instant t for which the curve has been constructed. If we wish to follow the changes which take place in time at a given point Xf we may either imagine the ordinates to be altered in the ratio of the respective times, or we may imagine the point Q to approach in such a way that OQ varies inversely as t. The foregoing construction has the defect that it gives no indication of the relative amplitudes in different parts of the wave-system. For this purpose we may construct the art'^kx curve which gives the relation between ai as ordinate and k as abscissa. If we draw a line through the origin whose gradient is x, the phase due to a particular elementary wave- train, viz. at - hzy wiU be represented by the difference of the ordinates of the curve and 26—2 388 Surfa4ie Waves [chap, ix the straight line. This difference will be stationary when the tangent to the ourve is parallel %o the straight line, ».e. when tdafdk =x, as aheady found. It is farther evident that the phase-difference, for elementary trains> of slightly different wave-lengths, will vary ultimately as the square of the increment of k. Also that the range of values of k for which the phase is sensibly the same will be greater, and consequently the resulting disturbance will be more intense, the greater the vertical chord of curvature of the curve. This explains the occurrence of the quantity td^<r/dJ^ in the denominator of the formula (8). In the hydiodjmamical problem of Art. 238 we have* ^(*)=1, cr««(7*, (12) whence da/div' i?**"*, d^r/di* = - i^*t"*, d*<Tld]c» = |j*ifc~*. . .(13) Hence, from (7), k^gt*l^x\ a^gtl2xy (14) and therefore ah V(27r) x^ or, on rejecting the imaginary part, The quotient in (9) is found to be comparable with {2x/gfl)^, so that the approximation holds only for times and places such that ^gt* is large com- pared with X. These results are in agreement with the more complete iuTestigation of Art. 238. The case of Art. 239 can of course be treated in a similar manner. It appears from (15), or from the above geometrical construction (the curve being now a parabola as in Art. 236), that in the procession of waves at any instant the wave-length diminishes continuaDy from front to rear; and that the waves which pass any assigned point will have their wave-lengths continually diminishing f. 242. We may next calculate the effect of an arbitrary, but steady, application of pressure to the surface of a stream. We shall consider only the state of steady motion which, under the influence of dissipative forces^ ♦ The difficulty as to convergence in this case is met by the remark that the formula (9> of Art. 238 gives 1?=- ^=limj^^Q- / ef^ COB fft 008 kxdk, where y is negative before the limit. t For further applications reference may be made to Havelock, *'The Propagation of Waves in Dispersive Media. . . ,** Proc. Boy. 8oc. t, Ixxxi p. 398 (1908). 241-242J Surface-Disturhanoe of a Stream 389 however small, will ultimately establish itself*. The question is in the first instance treated directly ; a briefer method of obtaining the principal result is explained in Art. 248. It is to be noted that in the absence of dissipative forces, the problem is to a certain extent indeterminate, for we can always superpose an endless train of free waves of arbitrary amplitude, and of wave-length such that their velocity relative to the water is equal and opposite to that of the stream, in which case they will maintain a fixed position in space. To avoid this indet^rminateness, we may avail ourselves of an artifice due to Rayleigh, and assume that the deviation of any particle of the fluid from the state of uniform flow is resisted by a force proportional to the f dative velocity. This law of resistance does not profess to be altogether a natural one, but it serves to represent in a rough way the eflect of small dissipative forces ; and it has the great mathematical convenience that it does not interfere with the irrotational character of the motion. For if we write, in the equations of Art. 6, Z = — /Lt (u — c), Y = — g — fjiv, Z = — fiw, (1) where c denotes the velocity of the stream in the direction of x-positive, the method of Art. 33, when applied to a closed circuit, gives ( ^-f /Lt ] i {udx -{- vdy -{- wdz) == 0, (2) whence j{udx -f vdy -f wdz) = Cer*^^ (3) Hence the circulation in a circuit moving with the fluid, if once zero, is always zero. We now have ^ = const. -gy^-ii(cx-^<f>)- |j«, (4) P this being, in fact, the form assumed by Art. 21 (2) when we write ^ = 9y-lJ^ipX'^4) (5) in accordance with (1) above.* To calculate, in the first place, the effect of a simple-harmonic distribution of pressure we assume r = _ a; + pe^y sin Jte, t= -y + pe^y coskx (6) c c \ * The first steps of the following investigation are adapted from a paper by Rayleigh, **The Form of Standing Waves on the Surface of Running Water," Proc, Lond. McUh, Soc t. xv. p. 09 (1883) [Papers, t. ii. p. 258], being simplified by the omission, for the present, of all reference to Capillarity. The definite integrals involved are treated, however, in a somewhat more general^ manner, and the discussion of the results necessarily follows a different course. The problem had been treated by Popoff, "Solution d'un probl^me sur les ondes permantontes," Lumvilh (2), t. iii. p. 251 (1858); his analysis is correct, but regard is not had to the indeter- minate character of the problem (in the absence of friction), and the results are consequently not pushed to a practical interpretation. 390 Surface Waves [chap, ix The equation (4) becomes, on neglecting the square of A^, 2 = ... - gy -h j3e*>' (*c* cos ix -f ftc sin kx) (7) P This gives for the variable part of the pressure at the upper surface (0 « 0) Pq = pp{(kc^ — j) cos fee + /Ltc sin fer}, (8) which is equal to the real part of pp (kc^ — 9 — iy^) «***• If we equate the coefficient to 0, we may say that to the pressure 3>o = C?c«« (9) corresponds the surface-form 9f^-ic-^-iJL,(^^'^' (^«) where we have written k for j/c*, so that 2ir/#c is the wave-length of the free waves which could maintain their position against the flow of the stream. We have also put /li/c = /^i, for shortness. Hence, taking the real parts, we And that the surface-pressure j>o = Ccos kx (11) produces the wave-form 9^ -<=■ ''- T^.l'if m This shews that if /t be small the wave-crests will coincide in position with the maxima, and the troughs with the minima, of the applied pressure, when the wave-length is less than 27r/#c; whilst the reverse hdds in the opposite case. This is in accordance with a general principle. If we impress on everything a velocity — c parallel to a;, the result obtained by putting /ij = in (12) is seen to be a special case of Art. 168 (14). In the critical case of i = #c, we have ?/>y = - — • sin fer, tl3) shewing that the excess of pressure is now on the slopes which face down the stream. This explains roughly how a system of progressive waves may be maintained against our assumed dissipative forces by a properly adjusted distribution of pressure over their slopes. 243. The solution expressed by (12) may be generalized, in the first place by the addition of an arbitrary constant to x, and secondly by a sum- mation with respect to i. In this way we may construct the eflect of any arbitrary distribution of pressure, say Vo=f{x\ (14) with the help of Fourier's Theorem (Art. 238 (4)). 242-243] Surface-Disturbance of a Stream 391 We will suppose, in the first instance, that / (x) vanishes for all but infinitely small values of x, for which it becomes infinite in such a way that r f(x)dx^P', (15) J -oo this will give us the efEect of an integral pressure P concentrated on an infinitely narrow band of the surface at the origin. Replacing C in (12) by P/tr.Sk, and integrating with respect to k between the limits and oo, we obtain kP /"* (k — k) cos fee — ui sin fee ,, ,, ^. ^''y = vi 0—1^3^)*-+^ — ^^ (^^) If we put i" = A + »m, where k, m are taken to be the rectangular co-ordinates of a variable point in a plane, the properties of the expression (16) are contained in those of the complex integral (17) /i^* It is known that the value of this integral, taken round the boundary of any area which does not include the singular ppint (C=<^)> ^ zero. In the present case we have c = K+ifjLj^, where k. and fi^ are both positive. Let us first suppose that x is positive, and let us apply the above theorem to the region which is bounded externally by the line m=0 and by an infinite semicircle, described with the origin as centre on the side of this line for which m is positive, and internally by a small circle surrounding the point (k, fi|). The part of the integral due to the infinite semicircle obviously vanishes, and it is easily seen, putting {'-c=re^', that the part due to the small circle is if the direction of integration be chosen in accordance with the rule of Art. 32. We thus obtain _«A;-(ic+»/ii) Jo A;-(ic+t^,) which is equivalent to On the other hand, when x is negative we may take the integral (17) round the contour made up of the line m=0 and an infinite semicircle lying on the side for which m is negative. This gives the same result as before, with the omission of the term due to the singular point, which is now external to the contour. Thus, for x negative, Jo k-{K+tfXi) Jo k + {K+%fj^) An alternative form of the last term in (18) may be obtained by integrating round the contour made up of the negative portion of the axis of k, and the positive portion of 'the axis of m, together with an infinite quadrant. We thus find /"o gifex /•• e'*"* I , 7 r-T dk + / -: ; : — : tdm =0, which is equivalent to / j—i ^,dk= — ? r-dm (20) Jok+{K+%fA^) Jom~fli+%K 392 Surface Waves [chap, ix This is for x positive. In the case of x negative, we must take as our contour the negative portions of the axes of k^ m, and an infinite quadrant. This leads to / 7—7 ^-,dk= r-iw, (21) as the transformation of the second member of (19). In the foregoing argument fi^ is positive. The corresponding results for the integral dC (22) / e*^ C-(«c-»>i) are not required for our immediate purpose, but it will be convenient to state them for future reference. For x positive, we find / , / . , dk= -r-7 ^,dk= — = r-dm; (23) whilst, for X negative, I ^ / . , iifc=-2^te<<^-^^»+ , f . . dk Jo *-(«c-»fti) Jo *+(«-»/ »>l) = -2irie*^-^^' + ^dm (24) Jo W -/*! -IK The verification is left to the reader*. c"» If we take the real parts of the formulae (18), (20), and (19), (21), respectively, we obtain the results which f oUow. The formula (16) is equivalent, for x positive, to kP ^ Jo (A: + /c)2 + fjij^ ==-2.e-^^^sin.x+f (7'-^-);"7t > (25) Jo (m-/ti)« + #c2 ^ ' and, for x negative, to (m + fjLj) e*^* dm '3>'i (m + fjL^)^+K^ ^^^^ The interpretation of these results is simple. The first term of (25) represents a train of simple-harmonic waves, on the down-stream side of the origin, of wave-length ^nrc^jg, with amplitudes gradually diminishing according to the law c""**!*. The remaining part of the deformation of the free-surface, expressed by the definite integrals in (25) and (26), though very great for small values of x, diminishes very rapidly as x increases in absolute value, however small the value of the frictional coefl&cient ^ii. When III is infinitesimal, our results take the simpler forms -^ . V == -- 27r sin /ex + ,— - — kP ^ ./' * + #c = — 27r sin #ca; + — r = dm, (27) * For another treatment of these integrals, see Dirichlet, YwUjBwngen ueber d. Lehre v. d. einfachen u. mehr/achen beatimmten IntegrcUen (ed. Arendt), Braunschweig, 1904, p. 170. 243] Surfaee-Disturbafice of a Stream 393 for X positive, and -% . y = y— — d* = — r-; — ^ dm, (28) for X negative. The part of the disturbance of level which is represented by the definite integrals in these expressioniB is now symmetrical with respect to the origin, and diminishes constantly as the distance from the origin increases. When kx is moderately large we find, by usual methods, the semi-convergent expansion j w« + fc« K*x* #c*x* "^ #c«a;« ^ ' It appears that at a distance of about half a wave-length from the origin, on. the down-stream side, the simple-harmonic wave-profile is fully established. The definite integrals la (27) and (28) can be reduced to known functions as foUows. U we put (k + K)x=UfWe have, for x positive, Jo «+K J KX tt = - Oi o; COS jca; +(^ -Si ica?) sin koj, (30) where, in conformity with the usual notation, Ciu= - I duj Sitt=/ - — tftt (31) ;« « Jo u The functions Oi u and Si u have been tabulated by Glaisher*. It appears that as u increases from zero they tend very rapidly to their asymptotic values and ^tt, respectively. For small values of u we have It* tt* • • • U* tt* ^'«=«-3:3! + 5.5! • • • » where y is Euler*s constant '5772. . . . It is easily found from (25) and (26) that when /l^ is infinitesimal, the integral depression of the surface is I " ydx = ^, (33) exactly as if the fluid were at rest. * ** Tables of the Nomerical Values of the Sine-Integral, Ck>sine-Integral, and Exponential- Integral," Phil, Trans. 1870; abridgments are given by Dale and by Jahnke and Emde. The expression of the last integral in (27) in terms of the sine- and cosine-integrals was obtained, in a different manner from the above, by Schlomilch, "Sor I'int^ale d^finie / znT* «""**»" Orelle, t. xxxiii. (1846); see also De Morgan, Differential and Integral Calculus, London, 1842, p. 654, and Dirichlet, Vorlesungen, p. 208. 394 Surface Waves [chap, ix 244. The expressions (25), (26) and (27), (28) alike make the elevation infinite at the origin, but this difficulty disappears when the pressure, which we have supposed concentrated on a mathematical line of the surface, is diffused over a band of finite breadth. To calculate the effect of a distributed pressure Po=/(a') (34) it is only necessary to write a; — a for a? in (27) and (28), to replace P by / (a) 8a, and to integrate the resulting value of y with respect to a between the proper limits. It follows from known principles of the Integral Calculus that if f^ be finite the integrals will be finite for all values of x. ' In the case of a uniform pressure f^^ applied to the part of the surface extending from — oo to the origin, we easily find by integration of (25), for a: > 0, gpy=-1^,^^^^)^^-^—-^, (35) where /n^ ^^ ^^^^ P^^ = ^* Again, if the pressure j>o ^^ applied to the part of the surface extending from to + oo , we find, for x < 0,* From these results we can easily deduce the requisite formulae for the case of a uniform pressure acting on a band of finite breadth. The definite integi'al in (35) and (36) can be evaluated in terms of the functions Ciu, Si u ; thus in (35) r*e-*"*dm r^sinfeCj, ,- «. v r^- - /o^v K — =— — i = i dk = liTT — Si Kx) cos #ca; + Ci /ca: sm K*. . . (37) In this way the diagram on p. 395 was constructed; it represents the case where the band (AB) has a breadth #c~^, or *159 of the length of a standing wave. The circumstances in any such case might be realized approximately by dipping the edge of a slightly inclined board into the surface of a stream, except that the pressure on the wetted area of the board would not be uniform, but would diimhish from the central parts towards the edges. To secure a uniform pressure, the board would have to be curved towards the edges, to the shape of the portion of the wave-profile included between the points Ay B in the figure. It will be noticed that if the breadth of the band be an exact multiple of the wave-length (27r//c), we have zero elevation of the surface at a distance, on the down-stream as well as on the up-stream side of the source of disturbance. 244] Wave-Profile 395 396 Surface Waves [chap, ix The diagram shews certain peculiarities at the points A^ B due to the discontinuity in the applied pressure. A more natural representation of a local pressure is obtained if we assume y» = L-^ (38) We may write this in the form p Y p r"^ = -.i— !^ = - 6-»+*»«dt, (39) n ^ provided it is understood that, in the end, only the real part is to be retained. On reference to Art. 242 (9), (10) we see that the corresponding elevation of the free surface is given by = — ,- ^ dk (40) n J Q fc — K — tui (41) gpy = - / — ; .- dm (42) By the method of Art. 243, we find that this is equivalent, for 35 >0, to i^«^ = ^ 27rte('+WU-») + J dm and, for « < 0, to :P f^ g<md+m0 Hence, taking real parts, and putting /L4 » 0, we find o z> ^ft • , P r* m cos m6 — #c sin m6 ^^ , _ ^t gpy^- 2KPe-^^ sin #ca? + - j- — ^ «"•*• am, [x > 0], (43) #cP r*m cos m6 — #c sin m6 ^^ , r ^t (44) The factor e^* in the first term of (43) shews the effect of diflhising the pressure. It is easily proved that the values of y and dy/da> given by these formulae agree when a? = 0*. 245. If in the problem of Art. 242 we suppose the depth to be finite and equal to A, there will be, in the absence of dissipation, indeterminateness or not, according as the velocity c of the stream is less or greater than (ghy, the maximum wave- velocity for the given depth ; see Art. 229. The difficulty presented by the former case can be evaded by the introduction of small frictional forces; but it may be anticipated from the preceding investigation that the main effect of these will be to annul the elevation of the surface at a distance on the up-stream side of the region of disturbed pressure, * A difFerent treatment of the problem of Art8. 243, 244 is given in a paper by Kelvin, " Deep Water Ship- Waves," Proc. R. S. Edin, t. xxv. p. 662 (1906) [Papers, t. iv. p. 368]. 244-245] Stream of Finite Depth 397 and if we assume this at the outset we need not complicate our equations by retaining the fiictional terms*. • For the case of a simple-harmonic distribution of pressure we assume ? as — x 4- jS cosh lc(y + h) sin fcc, JL := — y ^ j8 ginh i (y _|- J) cOg J^x^ (1) c as in Art. 231 (3). Hence, at the surface y » j3 sinh M cos Axr, (2) we have 2? = -jy - |(j2 « c«) = jS (*c« cosh *A - J sinh A*) cos ibr, . .(3) so that to the imposed pressure yo «* cos fcr (4) will correspond the surface-form _ C sinh tA , ^ " p' hc^ cosh kh — g sinh kh ^ ' As in Art. 242, the pressure is greatest over the troughs, and least over the crests, of the waves, or vice versd, according as the wave-length is greater or less than that corresponding to the velocity o, in accordance with general theory. The generalization of (5) by Fourier's method gives dk (6) I ^ sinh kh cos kx kc* cosh kh-g sinh kh as the representation of the effect of a pressure of integral amount P applied to a narrow band of the surface at the origin. This may be written irpC« _ r COS jXU/h) -p-'^-jo uoothu-gh/c^'^'' <^) Now consider the complex integral r«*» (8) / C coth ( - gh/c^ where {=:u +iv. The function under the integral sign has a singidar point at ^ = i^ too , according as a: is positive or negative, and the remaining singular points are given by the roots of T-gh ^®' Since (6) is an even function of x, it will be sufficient to take the case of x positive. * There is no difficulty in so modifying the investigaticn as to take the frictional forces into account, when these are very small. 398 Stirface Waves [chap, ix Let us first suppose that c^>gh. The roots of (9) are then all pure imaginaries ; viz. they are of the form ±%fi, where 3 is a root of tan 3 c* ,,^. V^T (^«> The smallest positive root of this lies between and ^ir, and the higher roots approximate with increasing closeness to the values (a + \) ir, where s is integral. We wiU denote these roots in order by ^o> ^» &t» • • • • Let us now take the integral (8) round the contour made up of the axis of u, an infinite semicircle on the positive side of this axis, and a series of small circles surrounding the singular points C=^^o> i^* ifit 'Hie part due to the infinite semicircle obviously vanishes. Again, it is known that if a be a simple root of / (C) =0 the value of the integral (C) I't /(f)* taken in the positive direction round a small circle enclosing the point ^=a is equal to* 2-fg "• (") Now in the case of (8) we have / ' (ti) = coth a -a (coth« a - I) =- l^ (l -^^\ + aA (12) whence, putting a =ifi„ the expression (11) takes the form 2»r5.«--^»*'* (13) where B,= i ' i (14) The theorem in question then gives / ^ r7-.<^^+ 4 r7-a<i^-2»rS.5.c-^'«'*=0 (16) J ^^uoothu-gh/c* J qU coth u-gh/c* o • If in the former integral we write - u for u, this becomes r_£2i(25/*) rf„=,S:B.«-^'.«/» (16) J ouoothu-gh/c^ ^ ' ^ ' The surface-form is then given by y=^..sr*.e-'-'* (17) It appears that the surface-elevation (which is symmetrical with respect to the origin) is insensible beyond a certain distance from the seat of disturbance. When, on the other hand, c^<gh, the equation (9) has a pair of real roots ( ±a, say), the lowest roots ( ±/3o) of ( 10) having now disappeared. The integral (7) is then indeterminate, owing to the function under the integral sign becoming infinite within the range of integration. One of its values, viz. the * principal value,' in Cauchy's sense, can however be found by the same method as before, provided we exclude the points f = ±a from the contour by drawing semicircles of small radius € round them, on the side for which v is positive. The parts of the complex integral (8) due to these semicircles will be * Forsyth, Theory of Functions, Art 24. 245-246] Stream of Finite Depth 399 where /^ (a) is given by (12) ; and their sum is therefore equal to 2irA sin "^ (18) « • • • where A- i , \ (19) , gh (gh \ The equation corresponding to (16) now takes the form SO that, if we take the principal value of the integral in (7), th^ surface-form on the side of X positive is y=-— ,^8in^ + ^,Sr5.e-^«'*. (21) Hence at a distance from the origin the deformation of the surface consists of the simple-harmonic train of waves indicated by- the first term, the wave-length 2irh/a being that corresponding to a velocity of propagation c relative to still water. Since the function (7) is symmetrical with respect to the origin, the corresponding result for negative values of rr is y=^.^sinf +|,S>.e«."» (22) The general solution of our indeterminate' problem is completed by adding to (21) and (22) terms of the form CcoB -T-+Dsm-T- (23) The practical solution, including the effect of infinitely small dissipative forces, is obtained by so adjusting these terms as to make the deformation of the surface insensible at a distance on the up-stream side. We thus get, finally, for positive values of x. y^-^^A^f^^XB.e-^-^ (24) and, for negative values of z. y=|iS«" *.«"•"'* (26) For a different method of reducing the definite integral in this problem we must refer to the paper by Kelvin cited below. 246. The same method can be employed to investigate the efiect on a uniform stream of slight inequalities in the bed*. Thus, in the case of a simple-harmonic corrugation given by y = — h + y cos kx, (1) ♦ Sir W. Thomson, "On Stationary Waves in Flowing Water,'* Phil, Mag. (6), t. xxu. pp. 353, 445, 517 (1886), and t. xxiii. p. 52 (1887) [Papers, t."iv; p. 270]. 400 Surface Waves [chap, ix the origin being as usual in the undisturbed surface, we assume ? = — a: 4- (a cosh hy -\- P sinh hy) sin fee, ' y (2) 1? = — y -f (a sinh hy •\- P cosh hy) cos fee. c The condition that (1) should be a stream-line is y = — a sinh kh-{- p cosh kh (3) The pressure-formula is 2 = const. — ^ + A»* (a cosh hy -{- fi sinh hy) cos fcr, (4) P approximately, and therefore along the stream-line ^ = ^ = const, -f {kc^a — gP) cos fcr, so that the condition for a free surface gives jfcc*a - jjS = (5) The equations (3) and (5) determine a and j3. The profile of the free surface is given by If the velocity of the stream be less than that of waves in still water of uniform depth h, of the same length as the corrugations, as determined by Art. 229 (4), the denominator is negative, so that the undulations of the free surface are inverted relatively to those of the bed. In the opposite case, the undulations of the surface follow those of the bed, but with a difierent vertical scale. When c has precisely the value given by Art. 229 (4), the solution fails, as we should expect, through the vanishing of the denominator. To obtain an intelligible result in this case we should be compelled to take special account of dissipative forces. The above solution may be generalized, by Fourier's Theorem, so as to apply to the case where the inequalities of the bed foUow any arbitrary law. Thus, if the profile of the bed be given by y=-h+f{x)=-k+- rdkT /(f) COS fc (a: -f) if, (7) n J J _oo that of the free surface will be obtained by superposition of terms of the type (6) due to the vajriouB elements of the Fourier-integral; thus f{^)coek{x-i) ^ g. cosh kh - g/kc^ . sinh kh In the case of a single isolated inequality at the point of the bed vertically beneath the origin, this reduces to _G r* cos fee „ n ] cosh kh -g/kc^ . sinh kh u cos (xu/h) =- r «^* r ^ J J — 00 vh J U cosh u - gh/c* . sinh u du, (9) 246-247] Inequalities in the Bed of a Stream 401 where Q represents the area included by the profile of the inequality above the general level of the bed. For a depression Q will of course be negative. The discussion of the integral / f cosh f - gh/c^ , sinh f can be conducted exactly as in Art. 245. The function to be integrated differs only by the factor ^/(sinh {) ; the singular points therefore are the same as before, and we can at once write down the results. Thus when c^>gh we find, for the surface-form, the upper or the lower sign being taken according as 2; is positive or negative. When c^<ghy the * practical' solution is, for x positive, j,=_^4^^8Hif +«srs.^«-'''«'» (12) ^ h sinha ^ A 1 'smjy, ^ ' and, for X negative, ^=1 ^1" ^'SS^ *^**'* (^^^ The symbols a, /3«, A^ B^ have here exactly the same meanings as in Art. 246*. 247. We may calculate, in a somewbat similar manner, the disturbance produced in the flow of a uniform stream by a submerged cylindrical obstacle whose radius h is small compared with the depth/ of its axisf. The cylinder is supposed placed horizontally athwart the stream. We write ^ = -«c(l+^^)+X, (1) where c denotes as before the general velocity of the stream, and r denotes distance from the axis of the cylinder, viz. r = V{a;* + (y+/)'} (2) the origin being in the undisturbed level of the surface, vertically above the axis. This makes 3<^/3r = f or r = 6, provided x ^® negligible in the neigh- bourhood of the cylinder. We assume /•OO X = I a{k) e*«' sin kcdh, (3) J * A very interesting drawing of the wave-profile produced by an isolated inequality in the bed is given in Kelvin's paper, Phil. Mag. (5), t. xxii. p. 520 [Paperst t. iv. p. 296]. The case of an abrupt change of level in the bed is discussed by Wien, Hydrodynamikf p. 201. The effect of inequalities of various kinds has been investigated by Qsotti in recent volumes of the RcTid. della r. Accad, dei Lincei, on the supposition that c' is so large in comparison with gh that the infiuence of gravity may be neglected. The problems are accordingly of the type considered in Arts. 73 . . . supra. t The investigation is taken from a paper "On some cases of Wave-Motion on Deep Water," Ann. di matematica (3). t. xxi. p. 237 (1913). I find that the problem had been suggested by Kelvin, Papers, t. iv. p. 369 (1904). L. H. 26 402 Surface Waves [chap, ix where a (A;) is a function of i, to be determined. For the equation of the free surface, assumed to be steady, we put 7] = I j3(i) cos kxdk (4) J The geometrical condition to be satisfied at the free surface is dy ""dx' <^^ wherein we may put y = 0. Since (1) is equivalent to ^ = - c» - 6«c e-*<>'+/' sin kxdk 4- x» (6) J for positive values of y +/, this condition is satisfied if ft2ce-»/ 4- a(k) = cj3(Jfe) (7) Again, the variable part of the pressure at the free surface is given by ?-»-4(l)' • 00 = — jiy — |c* — 6*c* I c~*' cos kxkdk + c J ^ dx /•oo roo = — jiy — ^c* — 6*c* / e-*' cos kxkdk -f- c 1 a (A) cos kxkdk, (8) where terms of the second order in the disturbance have been omitted. This expression will be independent of x provided gp(k) 4- Jfc6*c«e-*^ - kca{k) = (9) Combined with (7), this gives a(*) = |i^6«ce-*/, ^(A) = 2^^. (10) where K = g/c^ (11) as in Art. 242. Hence = 26» I J I * fe-*^ COS fcrd* ^f. * "" jl^-^*' <'2> The integral is indeterminate, but if a; be positive its principal value is equal to the real part of the expression /« p—imf—mx -.- dm (13) tm — K 247-248] Waves dtte to a Submerged Cplinder 403 Adopting this we have 71 = ,- -J,!, — 27ric6*e"*^ sin kx ' «*+/* o r« /"" C*^ sin wif — m cos m/) e-*»* , ,. .. For large values of x the second term is alone sensible. Since the value of 77 in (12) is an even function of x we must have, for X negative, 26*/ « !• * . « ,0 r*(Ksinm/— mcosm/*)e'** , ,--. 17 = -5—=^ + ^Kb^e"^ sm ica; - 2#c62 ^ =^—5- — ^-^^ dm. ... (15) ' x^-\'P Jo m* + ic* On the disturbances represented by these formulae we can superpose any system of stationary waves of length 27r/#c, since these could maintain their position in space, in spite of the motion of the stream ; and if we choose as our additional system 7) = — 27^06*6"*^ sin #cx (16) we shall annul the disturbance at a distance on the up-stream side (x < 0), as is required for a physical solution. The result is 26*/ \ 17 = g *^^g — 4jr#c6*e"*^ sin kx -f- &c. [x > 0], ^ ^^ [ (17) ^ = a;2 ^ ya -^ ^^- [a;<0]. It appears that there is a local disturbance immediately above the obstacle, followed by a train of waves of length 27rc^/g on the down-stream side*. 248. If in the problems of Arts. 243, 245 we impress on everything a velocity — c parallel to x, we get the case of a pressure-disturbance advancing with constant velocity c over the surface of otherwise still water. In this form of the question it is not difficult to understand, in a general way, the origin of the train of waves following the disturbance. If, for example, equal infinitesimal impulses be applied in succession to a series of infinitely close equidistant parallel lines of the surface, at equal intervals of time, each impulse will produce on its own account a system of waves of the character investigated in Art. 239. The systems due to the different impulses will be superposed, with the obvious result that the only parts which reinforce one another will be those which have the wave-length appropriate to the velocity c with which the disturbing influence advances over the surface, and which are (moreover) travelling in the direction of this advance. And the investigations of Arts. 236, 237 shew that the groups of * If we investigate the asymptotic expansion of the definite integral in (13), when k/ is large, we find on substitution in (12) that the most important term gives -2b'^/l{^+P), and so cancels the first term in the above values of ri. 2ft— 2 404 Surface Waves [OHAP. IX waves, of this particular length, which are produced, are continually being left behind. This question can be treated from a general standpoint by an application of Kelvin's method (Art. 241). Let us suppose that the disturbing influence, imagined to be concentrated in a line perpendicular to the axis of x, is advancing with the constant velocity c in the direction of x negative. Let be its position at the instant under consideration, and let t denote the time that has elapsed since it occupied Q any previous position Q, so that OQ = ct. It is required to find the actual disturbance at any assigned point P. We write X = OPy $ = PQ = ct — X (1) Taking the formula (8) of Art. 241 as a basis, we have for that part of the disturbance at P which was originated at Q an expression of the type where k is now to be regarded as a function of t determined by the relation tp^^i=^ct-x, (3) and m = V i \td^<Tldk^ \ (4) The relation between a and k will depend of course on the dynamics of the particular kind of waves we are considering. We have to integrate (2) with respect to t between the limits and oo , but only those elements of the integral are important for which the exponential is nearly stationary in value. Now, having regard to (3), 248] Waves diie to a Travelling Disturbance 405 whilst by differentiation of (3) d^a dk da ___ ^ ^Wdi'^dk^^"^' Hence, writing V for the group- velocity {da/dk), we have ^ (of - if) = cr - Ac, ^, (of - k$) = ^ ^ ^^' , (5) where the sign is the opposite to that of d^a/dk*. In order that the phase at P, viz. at — k^± Jtt, may be stationary as regards variation in the position of Q we must have, then, a=kc (6) This determines k, and the corresponding value of t then follows from (3). If we denote these special values of k and t by k and r, respectively, the corresponding value of at — k^ will be ar — K^ = k{ct — i)= KX, by (6). Hence if we write k^K-hk', t = T + t\ (7) the index in (2) will take the form approximately. Since " c+»«(«^-<'W'«'(fe' = -'^T''^, (8) -flo \U — c\ by Art. 241 (4), we obtain finally, for the disturbance at the point x, the simple expression >? = |^L^e- (9) It will be understood that U here denotes the value of the group-velocity corresponding to k = k. It appears from (6) that the wave-length of the progressive wave-train represented by this formula is that of a free wave-train whose velocity of propagation is c. Also, since by (3) x = {C'-'U)t, (10) the values of a; to which the preceding calculation applies will be positive or negative according as J7 $ c. If ?7 < c, as is the case of gravity waves on a liquid, the train follows the initiating disturbance, whilst if 17 > c, as is the case of capillary waves (Art. 266), it precedes it. If there is more than one value of k satisfying (6) there will be a term in 7) corresponding to each of these. / 406 Surface Waves [chap, ix Referring now to Art. 239 (33), we see that to find the elevation 17 in the case of waves on deep water due to a travelling pressure we must put <f> (k) ^ iaP/gp (11) Since U is now = Jc, we obtain, on taking the real part, 77 = sinKX. (12) 9P in agreement with (27) of Art. 243*. As the preceding investigation involves a double approximation, it may be worth while to give another method of arriving at the result (9) which will indicate very readily the condition under which it holds. If we introduce the hypothesis of a small frictional force varying as the velocity, the formula (6) of Art. 241, when modified so as to apply to the case of a travelling disturbance, takes the form TJ = ~r j r<^ (k) 6<<r*-i*(ct-a:) dJc ^ r ^Urt+mct^a^) dk\ 6'^^^ dt. . . (13) The integration with respect to t gives "^ "" 27rj i/i - t (a-~kc) ^27tJ J/i -> i (a -h kc) ^^^^ The quantity /x is by hypothesis small, and will in the limit be made to vanish. The most important part of the result will therefore be due to values of k in the first integral which make a=kc (15) approximately. Writing as before k = k-{- k\ where #c is a root of this equation, we have --kc=(^-c)k'=={U-c)kf, (16) nearly, where U denotes the group- velocity corresponding to the wave-length 2n/K. The important part of (14) is therefore Now if a be positive we havet p e<"»* dm ^ f27re-««, [x > 0] J^aoa + im [ 0, [a;<0] ^ ' * This method of obtaining the formula (12) was indicated in a footnote on p. 416 of the preceding edition. t The results quoted are equivalent to the familiar formulae mxdin a /"* cos mxdm _ . f* m sin w wi* =ire*«« (where the upper or lower sign is to be chosen according as a; is positive or negative), but can be obtained directly by a contour integration. 248-249] Waves dtie to a Travelling Disturbance 407 whilst Hence ii U < c p e*"^ dm _ J ^oo a — im 0, [X > 0] 27re«*. [a;<0] ^ ^ ^^ <f>(K) e^' ^,|^;^(„C7)^ or 0, (20) C-I7 according as a; % ; whilst it U > c ^ = 0, or ^M^%-i^«/(cr-e) (21) ' U — c in the respective cases. If we now make /i -^ we obtain our former results. The approximation in (16) is valid only if the quotient d^a/dk^ .k' -^(U-c) (22) is small even when k'x is a moderate multiple of 27r. This requires that d^a/dk^ H- (£7 - c) a; (23) should be small. Unless V = c, exactly, the condition is always fulfilled if X be sufficiently great. It may be added that the results (20), (21) are accurate, in the sense that they give the leading term in the evaluation of (14) by Cauchy's method of residues. Cf. Art. 242. 249. The preceding results have a bearing on the question of 'wave- resistance.' Taking for definiteness the case of Z7 < c, let us imagine a fixed vertical plane to be drawn in the rear of the disturbing agency. If E be the mean energy of the waves, the space in front of this plane gains, per unit time, the additional energy cE, whilst the energy transmitted across the plane is VE, by Art. 237. Hence if R be the resistance experienced by the disturbing body r^^hIe (1) Ii U > Cf the fixed plane must be taken in advance, and the result is R = ^^::-^E (2) Thus, in the case of a disturbance advancing with velocity c [< ^/{gh)'\ over still water of depth A, we find, on reference to Art. 237, R = *"«-(■ -iJir*)- (') where a is the amplitude of the waves. As c increases from to ^/(gh), kK diminishes from oo to 0, so that R diminishes from \gpa^ to 0. When c>\/(gh\ the effect is merely local, and /? = 0*. It must be remarked, ♦ Cf. Sir W. Thomson, "On Ship Waves," Proc. Inst. Mech. Eng. Aug. 3, 1887 [Popular Lectures and Addresses, London, 1889-94, t. iii. p. 450]. A foimula equivalent to (3) was given in a paper by the same author, Phil. Mag. (5), t. xxii. p. 451 [Pajfersj t. iv. p. 279]. 408 Surfdce Waves [chap, ix however, that the amplitude a due to a disturbance of given type will also vary with c. For instance, in the case of Art. 244 (43), a oc Ke-^^y where K = ^/c*, the depth being infinite. Hence R a c-*e-*»^/«' (4) An interesting variation of the general question is presented when we have a layer of one fluid on the top of another of somewhat greater density. If p, p' be the densities of the lower and upper fluids, respectively, and if the depth of the upper layer be A', whilst that of the lower fluid is practically infinite, the results of Stokes quoted in Art. 232 shew that two wave-systems may be generated, whose lengths {^Jk) are related to the velocity c of the disturbance by the formulae k' ^ " pcothKh' + p''k ^^^ It is easily proved that the value of k determined by the second equation is real only if c2<P:^gh^ (6) P If c exceeds the critical value thus indicated, only one type of waves will be generated, and if the difference of densities be slight the resistance will be practically the same as in the case of a single fluid. But if c fall below the critical value, a second type of waves may be produced, in which the amplitude at the common boundary greatly exceeds that at the upper surface; and it is to these waves that the * dead-water resistance' referred to in Art. 232 is attributed*. The problem of the submerged cylinder (Art. 247) furnishes an instance where the wave-resistance to the motion of a solid can be calculated. The mean energy, per unit area of the water surface, of the waves represented by the second term in equation (14) of that Art. is E = ^p {i^KbH-^f)^. Since U = Jc, we have from (1) R = 4^^gpb^K^e-^f (7) For a given depth (/) of immersion, this is greatest when k/= 1, or c = V{9f) (8) In terms of the velocity c we have R = i^^g^ph* . c-«e-«'^/^' (9) The graph of 12 as a function of c is appended!. * Ekman, l.c. ante p. 361. t Ann, di mat.. I.e. The same law of resistance as a function of the velocity c has heen obtained by Havelock, in the case of various types of surface disturbance, "Ship Resistance. . .." Proc. R. 8. t. Ixxziz. p. 489 (1913). A previous paper by the same author on "The Wave-Making Resistance of Ships, . . .," Proc. R. S. t. Ixxxii. p. 276 (1909), may also be referred to. 249-250] Wave-Hesistance 409 ^fgf) Waves of Finite Amplitude. 250. The restriction to "infinitely small' motions, in the investigations of Arts. 227, . . . implies that the ratio (a/A) of the maximum elevation to the wave-length must be small. The determination of the wave-forms which satisfy the conditions of uniform propagation without change of type, when this restriction is abandoned, forms the subject of a classical research by Stokes*. The problem is most conveniently treated as one of steady motion. If we neglect small quantities of the order a'/A*, the solution of the problem in the case of infinite depth is contained in the formulae I - = — X -f j3e*^ sin fee. ?- r= — y + )3e*v cos kx (1) The equation of the wave-profile = is found by successive approxi- mations to be y = j3e*v cos A:x = j3 (1 + % + \k^y^ + . . . ) cos fee = i*^2 + jS (1 + gPjS2) cos kx + \kp^ cos2fer + p«j3» cosSfec + . . . ; . .(2) or, if we put jS (1 + %h^p^) = a, y — \ka^ = a cos kx + \ka^ cos 2fei; + fi^a' cos Sfec + (3) * "On the theory of Osoillatoiy Waves," CavA, Trans, t. yiii. (1847) [reprinted, with a "Supplement," Papers, t. i. pp. 197, 314]. The outlines of a more general investigation, including the case of permanent waves on the common surface of two horizontal currents, have been given by Helmholtz, *'Zur Theorie von Wind und Wellen," Bed. MonaUber. July 25, 1889 [Wiss. Ahh. t. iii. p. 309]. See abo Wien, Hydrodynamik, p. 169. t Rayleigh, l.c, ante p. 252. 410 Surface Waves [chap, ix So far as we have developed it, this coincides with the equation of a trochoid, in which the circumference of the rolhng circle is %v\hy or A, and the length of the arm of the tracing point is a. We have still to shew that the condition of uniform pressure along this stream-Une can be satisfied by a suitably chosen value of c. We have, from (1), without approximation ^ = const, -gy- \c^ {1 - 2ij86*»' cos ht + PjS^e**''}, (4) and therefore, at points of the line y — jSe*" cos fee, ^ = const. ^{h?-g)y- p^c^jS^e^kv = const. + (*c* - 5^ - Pc2j8«) y + (5) Hence the condition for a free surface is satisfied, to the present order of approximation, provided c2 = 1+ Pc2j82 = |(1 + A:«a2) (6) This determines the velocity of progressive waves of permanent type, and shews that it increases somewhat with the amplitude a. For methods of proceeding to a higher approximation, and for the treatment of the case of finite depth, we must refer to the original in- vestigations of Stokes*. The figure shews the wave-profile, as given by (3), in the case of fci = J, or a/A = -0796. The approximately trochoidal form gives an outUne which is sharper near the crests, and flatter in the troughs, than in the case of the simple-harmonic waves of infinitely small amplitude investigated in Art. 229, and these features become accentuated as the amplitude is increased. If the trochoidal form were exact, instead of merely approximate, the Umiting form would have cusps at the crests, as in the case of Gerstner's waves to be considered presently. In the actual problem, which is one of irrotational motion, the extreme form has been shewn by Stokes f, in a very simple manner, to have sharp * See also Rayleigh, PhU, Mag. (6), t. xxi. p. 183 (1911), and Pwc, Boy. 8oc. A, t. xci. p. 345 (1915). The latter paper includes the case of standing waves. Reference may be made also to Priestley, Camb. Proc. t. xv. p. 297 (1909), and Wilton, PhU. Mag. (6), t. xxvii. p. 385 (1914). t Papers, t. i. p. 227. 250] Finite Waves of Permanent Type 411 angles of 120°. The question being still treated as one of steady motion, the motion near the angle will be given by the formulae of Art. 63 ; viz. if we introduce polar co-ordinates r, with the crest as origin, and the initial line of 6 drawn vertically downwards, we have = Of" cos md, (7) with the condition that = when fl = ± a (say), so that ma = Jtt. This formula leads to g = wCr«-i, (8) where q is the resultant fluid-velocity. But since the velocity vanishes at the crest, its value at a neighbouring point of the free surface will be given by q^ = 2gr cos a, (9) as in Art. 24 (2). Comparing (8) and (9), we see that we must have m = f , and therefore a = Jtt*. In the case of progressive waves advancing over still water, the particles at the crests, when these have their extreme forms, are moving forwards with exactly the velocity of the wave. Another point of interest in connection with these waves of permanent type is that they possess, relatively to the undisturbed water, a certain momentum in the direction of wave-propagation. The momentum, per wave- length, of the fluid contained between the free surface and a depth h (beneath the level of the origin) which we will suppose to be great compared with A, is - p I p^ dxdy = pchX, (10) since ^ = 0, by hypothesis, at the surface, and = ch, by (1), at the great depth h. In the absence of waves, the equation to the upper surface would be y = JA»^ by (3), and the corresponding value of the momentum would therefore be pc(A + iJfca»)A (11) The difference of these results is equal to Trpa^Cy (12) which gives therefore the momentum, per wave-length, of a system of progressive waves of permanent type, moving over water which is at rest at a great depth. * The wave-profile has been investigated and traced by Michell, "The Highest Waves in Water," Phil, Mag. (5), t. xxxvi. p. 430 (1893). He finds that the extreme height is -142 X, and that the wave- velocity is greater than in the case of infinitely small height in the ratio of 1*2 to 1. See also Wilton, PhU, Mag. (6), t. zxvi p. 1053 (1913). 412 Surface Waves [chap, ix To find the vertical distribution of this momentum, we remark that the equation of a stream-line ^ = cV is found from (2) by writing y '\-h' for y, and jSe"-**' for j3. The mean-level of this stream-line is therefore given by y = - A' -I- J*^%-»*' (13) Hence the momentum, in the case of undisturbed flow, of the stratum of fluid included between the surface and the stream-line in question would be, per wave-length, />cA{A' + iA^«(l-«-"^)} (14) The actual momentum 1>eing pch'\ we have, for the momentum of the same stratum in the case of waves advancing over still water, -npa^c (1 - e-»*') (15) It appears therefore that the motion of the individual particles, in these progressive waves of permanent tjrpe, is not purely oscillatory, and that there is, on the whole, a slow but continued advance in the direction of wave- propagation*. The rate of this flow at a depth K is found approximately by differentiating (15) with respect to h\ and dividing by pA, viz. it is Jfc«a«ce-»*' (16) This diminishes rapidly from the surface downwards. 251. A system of eocfui, equations, expressing a possible form of wave- motion when the depth of the fluid is infinite, was given so long ago as 1802 by Gerstnerf, and at a later period independentiy by Bankine]:. The circumstance, however, that the motion in these waves is not irrotational detracts somewhat from the physical interest of the results. If the axis of x be horizontal, and that of y be drawn vertically upwards, the formulae in question may be written a; = a 4- T e** sin i (a -I- ci), y = 6 — t c** cos i (a + ci), (1) where the specification is on the Lagrangian plan (Art. 16), viz. a, 6 are two parameters serving to identify a particle, and x, y are the co-ordinates of this particle at time t. The constant h determines the wave-length, and c is the velocity of the waves, which are travelling in the direction of x-negative. To verify this solution, and to determine the value of c, we remark, in the first place, that iti-'-«-. <^) * Stokee, {.e. ante p. 400. Another very simple proof of this statement has heen given by Rayleigh, ^e. atUe p. 252. t Professor of Mathematics at Prague, 1789-1823. His paper, "Theorie der Wellen," was published in the Ahh. d. k, bdhm. Oes. d, Wiss. 1802 [Qilbert's AnnaUn d. Physik, t. xzxit (1809)]. X "On the Exact Form of Waves near the Surface of Deep Water/' PhU, Trans. 1863 [Papers, p. 481]. I ,.(3) 280-261] Oerstner'a Waves 413 so that the Lagrangian equation of continiiity (Art. 16 (2)) JB satisfied. Again, sabstitatijig from (1) in the equations of motion (Art. 13), we find a(f + ^)" fe's"*™ *(« + «). A (- + w) *"'e" COS i (o + «) + fci'e"" ; whence ^ = const. - 5 Jfc - T e** cos A: (a + cf)[ - c*c" cos k(a+ ct) + Jc»e"». W For a particle on the free aurface the pieesure mnst be constant; this requires "■-I c^' as in Art. 229. This makes ^ = cODBt. '-gb + ic»fi»» (6) It is obvious from (1) that the path of any particle (a, &) is a circle of radios ir>e**. The figure shews the forms of the lines of equal pressure b = const., for a series of equidistant values of b*. These curves are trochoids, obtained by * The diagiam U very simil&r to the one giTen originally bj Qentner, and copied more or ess closely by mbsequent writen. A version of Qeratner's investigation, inoluding in imi» respect a correction, wM given in the seoond edition of thi« work. Art. 233. 414 Surface Waves [chap, ix rolling circles of radii Ic^ on the under sides of the lines y = 6 4- A~"^, the distances of the tracing points from the respective centres being Ar^e**. Any one of these lines may be taken as representing the free surface, the extreme admissible form being that of the cycloid. The dotted lines represent the successive forms taken by a line of particles which is vertical when it passes through a crest or a trough. It has already been stated that the motion of the fluid in these waves is rotational. To prove this, we remark that = ISfe** sin i (a + c<)} -f- ce«* Sa, ' (7) which is not an exact differential. The circulation in the boundary of the parallelogram whose vertices coincide with the particles {a,yh\ (a + 8a, 6), (a, 6 -f- 86), {a + 8o, 6 4- 86) is, by (7), - ^ (ce^** 8a) 86, and the area of the circuit is Ij^ 8a86 = (1 - e"*) 8a86. 3 (a, 6) Hence the vorticity (co) of the element (a, 6) is 2kce^ ^ = -i3^*b (8) This is greatest at the surface, and diminishes rapidly with increasing depth. Its sense is opposite to that of the revolution of the particles in their circular orbits. A system of waves of the present type cannot therefore be originated from rest, or destroyed, by the action of forces of the kind contemplated in the general theorem of Arts. 17, 33. We may however suppose that by properly adjusted pressures applied to the surface of the waves the liquid is gradually reduced to a state of flow in horizontal lines, in which the velocity (m') is a function of the ordinate (y') only*. In this state we shall have dx'jda == 1, while y' is a function of 6 determined by the condition 9 («', y') d (x, y) d (a, 6) a (a, 6) ' (9) or g6 ='~^*** ^^^^ * For a fuller statement of the aigument see Stokes' PaperSj t. i. p. 222. 251-252] Gerstner's Waves 415 ■^"■""^ ■%-%%--^%-^'-"- <"> and therefore u' = ce^w (12) Hence, for the genesis of the waves by ordinary forces, we require as a foundation an initial horizontal motion, in the direction opposite to that of propagation of the waves ultimately set up, which diminishes rapidly from the surface downwards, according to the law (12), where 6 is a function of y' determined by y' = 6 - Ik-^e^^ (13) It is to be noted that these rotational waves, when established, have zero momentum. 252. Scott Russell, in his interesting experimental investigations*, was led to pay great attention to a particular type which he called the 'solitary wave.' This is a wave consisting of a single elevation, of height not necessarily small compared with the depth of the fluid, which, if properly started, may travel for a considerable distance along a uniform canal, with little or no change of type. Waves of depression, of similar relative amplitude, were found not to possess the same character of permanence, but to break up into series of shorter waves. Russell's * solitary ' type may be regarded as an extreme case of Stokes' oscillatory waves of permanent type, the wave-length being great compared with the depth of the canal, so that the widely separated elevations are practically independent of one another. The methods of approximation employed by Stokes become, however, unsuitable when the wave-length much exceeds the depth; and subsequent investigations of solitary waves of permanent type have proceeded on different lines. The first of these was given independently by Boussinesq| and Rayleigh J. The latter writer, treating the problem as one of steady motion, starts virtually from the formula <!>']- uls = F(X'hiy)=^ e^^ F (x), (1) where F {x) is real. This is especially appropriate to cases, such as the present, where one of the family of stream-lines is straight. We derive from (1) <^ = l'-|-jl'" + |^Fv-..., = yl?''-|jF"-f.|^J^-...,..(2) where the accents denote differentiations with respect to x. The stream-line = here forms the bed of the canal, whilst at the free surface we have ♦ "Report on Waves," BriL Ass. Rep. 1844. t Comptes RenduSf June 19, 1871. t Ic. ante p. 252. 416 ^rface Waves [chap, ix ^ = — ch, where c is the uniform velocity, and h the depth, in the parts of the fluid at a distance from the wave, whether in front or behind. The condition of uniform pressure along the free surface gives w* + V* = c« - 2sr (y - A), (3) or, substituting from (2), J'2 _ y2j/j''' ^ y2j''2 + _ . = c« - 2<7 (y - A) (4) But, from (2) we have, along the same surface, y^'-|^^'"+ ... = -cA (5) It remains to eliminate F between (4) and (5) ; the result will be a differential equation to determine the ordinate y of the free surface. If (as we will suppose) the function F' (x) and its differential coefficients vary so slowly with X that they change only by a small fraction of their values when x increases by an amount comparable with the depth A, the terms in (4) and (5) will be of gradually diminishing magnitude, and the elimination in question can be carried out by a process of successive approximation. Thus, from (5) and if we retain only terms up to the order last written, the equation (4) becomes ■ or, on reduction, 1 2y" ly'*_ 1 igjy-h) y* S y 3 y» h* c*h* ^ ' If we multiply by y", and integrate, determining the arbitrary constant so as to make y' = ioi y = h, we obtain 1 ly'» 1 y-h gjy-hy y S y A "^ A» c»A» ' or /• = 3^X^-f) («) Hence y' vanishes only for y = A and y = c^/g, and since the last factor must be positive, it appears that c*/g is a maximum value of y. Hence the wave is necessarily one of elevation only, and, denoting by a the maximum height above the undisturbed level, we have c^^g{h + a) (9) which is exactly the empirical formula for the wave-velocity adopted by Russell. 252] The Solitary Wave 417 The extreme form of the wave must, as in Art. 250, have a sharp crest of 120° ; and since the fluid is there at rest we shall have c* = 2ga. If the formula (9) were applicable to such an extreme case, it would follow that a = A. If we put, for shortness, (10) we find, from (8), X (11) (12) v=±i(i-5)*. the integral of which is 71 = a sech* \ ^ if the origin of x be taken beneath the summit. There is no definite 'length ' of the wave, but we may note, as a rough indication of its extent, that the elevation has one-tenth of its maximum value when xjh = 3'636. The annexed drawing of the curve y = 1 + ^sech*|a; represents the wave-profile in the case a = \Ji. For lower waves the scale of y must be contracted, and that of x enlarged, as indicated by the annexed table giving the ratio 6/A, which determines the horizontal scale, for various values of ajh. It will be found, on reviewing the above investigation, that the approximations consist in neglecting the fourth power of the ratio (A + a)/26. If we impress on the fluid a velocity — c parallel to x we get the case of a progressive wave on still water. It is not difficult to shew that, if the ratio ajh be small, the path of each particle is then an arc of a parabola having its axis vertical and apex upwards*. It might appear, at first sight, that the above theory is inconsistent with the results of Art. 187, where it was argued that a wave of * Boussinesq, /.c. ajh •1 hjh 1-916 •2 ' 1-414 •3 1 1-202 •4 1-080 •6 1000 •6 -943 •7 •900 •8 .866 -9 -839 1-0 •816 L. H. 27 418 Surface Waves [chap, ix finite height whose length is great compared with the depth must inevitably suffer a continual change of form as it advances, the changes being the more rapid the greater the elevation above the undisturbed level. The in- vestigation referred to postulates, however, a length so great that the vertical acceleration may be neglected, with the result that the horizontal velocity is sensibly uniform from top to bottom (Art. 169). The numerical table above given shews, on the other hand, that the longer the 'solitary wave ' is, the lower it is. In other words, the more nearly it approaches to the character of a * long ' wave, in the sense of Art. 169, the more easily is the change of type averted by a slight adjustment of the particle- velocities*. The motion at the outskirts of the solitary wave can be represented by a very simple formula. Considering a progressive wave travelling in the direction of x- positive, and taking the origin in the bottom of the canal, at a point in the front part of the wave, we assume <t> =^e-'»(*-«') cos my (13) fhis satisfies v'<^ =0.. and the surface-condition |t^^|=« (^*) will also be satisfied for ?/ = ^, provided , -tanm^ ^ . '"-^^-mh- <^'^' This will be found to agree approximately with Rayleigh's investigation if we put m =6~^ The above remark, which was kindly communicated to the author by the late Sir George Stokesf , was suggested by an investigation by McCowanJ , who shewed that the formula ? — ^= -(x+iy) -fa tanh J m(a;-fty) (16) c satisfies the conditions very approximately, provided c« = ^ tan mA, (17) m and wki = f sin«m(A + |o), a =a tan Jm (A -fo), (18) where a denotes the maximum elevation above the mean level, and a is a subsidiary constant. In a subsequent paper § the extreme form of the wave when the crest has a sharp angle of 120° was examined. The limiting value of the ratio ajh was found to be -78, in which case the wave- velocity is given by c* = l-56flrA. 253. By a slight modification the investigation of Rayleigh and Boussinesq can be made to give the theory of a system of oscillatory waves of finite height in a canal of limited depth * Stokes. "On the Highest Wave of Uniform Propagation," Proc. Camb. Phil. Soc, t. iv. p. 361 (1883) [Papera, t. v. p. 140]. t Cf. Papers, t. v. p. 162. t "On the Solitary Wave," Phil. Mag. (5), t. xxxii. p. 46 (1891). § "On the Highest Wave of Permanent Tjrpe," PhU. Mag. (5), t. xxxviii. p. 361 (1894). II Korteweg and De Vries, "On the Change of Form of Long Waves advancing in a Rect- angular Canal, and on a New Type of Long Stationary Waves," Phil. Mag. (6), t. zxxix. p. 422 (1896). The method adopted by these writers is somewhat different. Moreover, as the title 252-253] Theory of Korteweg and De Vries 419 In the steady-motion form of the problem the momentum per wave-length (X) is repre- sented by j jpudxdy= -p j j ^^^y= -P^i^* (1^) where ^^ corresponds to the free surface. If A be the mean depth, this momentum may be equated to pch\, where c denotes (in a sense) the mean velocity of the stream. On this understanding we have, at the surface, V^i = - ch, as before. The arbitrary constant in (3), on the other hand, must be left for the moment undetermined, so that we write u^+v^=C-2gy, (20) We then find, in place of (8), tr=^t{y-l){hi-y){y-h,) (21) where k^ , ^2 ^^^ ^^^ upper and lower limits of y, and '=.^ : ^''^ It is implied that I cannot be greater than A,. If we now write y=^ cos* x + ^ s^* X> (2^) we find ^^ = V{1 -ifc*sin«x}, (24) '^Vl^*;^}' *'=*t^ ^^> Hence, if the origin of x be taken at a crest, we have and y=K +(^1 -^2) en* -. [mod. k] (27)* The wave-length is given by V(l-ik*sin*x) ^=2^^" ^Ti-fcaTx =2^^i W (28) Again, from (23) and (24), j\d.=2fffJ'^-^';'l^^.^^dx=mI'^{k)Hh^-l)EAk)} (29) Since this must be equal to AX, we have (h-l) Fi [k) =(^ -0 El (Jfc) (30) In equations (25), (28), (30) we have four relations connecting the six quantities ^, ^, /, k, X, fif so that if two of these be assigned the rest are analytically determinate. The wave- velocity c is then given by (22) f- For example, the form of the waves, and their velocity, are determined by the length X, and the height A^ of the crests above the bottom. The solitary wave of Art. 252 is included as a particular case. If we put l=h^, we have k = l, and the formulae (28) and (30) then shew that X = x , A, = A. indicates, the paper includes an examination of the manner in which the wave-profile is changing at any instant, if the conditions for permanency of type are not satisfied. For other 'modifications of Rayleigh*8 method reference may be made to Gwyther, Phil, Mag, (6), t. 1. pp. 213, 308, 349 (1900). * The waves represented by (27) are called *cnoidal waves* by the authors cited. For the method of proceeding to a higher approximation we must refer to the original paper. t When the depth is finite, a question arises as to what is meant exactly by the * velocity of propagation.' The velocity adopted in the text is that of the wave-profile relative to the centre of inertia of the mass of fluid included between two vertical planes at a distance apart equal to the wave-length. Cf. Stokes, Papers, t. i. p. 202. 27—2 420 Surface Waves [chap, ix 254. The theory of waves of permanent type has been brought into relation with general dynamical principles by Helmholtz*. If in the equations of motion of a 'gjrrostatic ' system, Art. 141 (24), we put where V is the potential energy, it appears that the conditions for steady motion, with q^, q^, ... g„ constant, are l(F + iC) = 0, ^(7 + X) = o, ..., ^(F + iC) = o,..(2) dqi dq2 oq^ where K is the energy of the motion corresponding to any given values of the co-ordinates q^, jj* • • • ?n when these are prevented from varying by the application of suitable extraneous forces. This energy is here supposed expressed in terms of the constant momenta corresponding to the ignored co-ordinates x* x'» • • • > *^^ ^^ ^^^ palpable co-ordinates Ji, y2> • • • ?n« I* ^^7 however also be expressed in terms of the velocities x> x'» • • • *^^ *^® co-ordinates 9], 92> • • • 9n > i^ ^^ form we denote it by Tq. It may be shewn, exactly as in Art. 142, that dTo/dqr = — dK/dq^ so that the conditions (2) are equivalent to |-(7-2'o) = 0, i^^(y-To) = 3^-^(7- To) = 0. ..(3) Hence the condition for free steady motion with any assigned constant values of ji, jj, ... j„ is that the corresponding value of 7 + JT, or of 7 — Tq* should be stationary. Cf. Art. 203 (11). Further, if in the equations of Art. 141 we write — dV/dqr + Qr for Q^y so that Qr now denotes a component of extraneous force, we find, on multiplying by ji, 92> • • • 9n i^ order, and adding, where "ST is the part of the energy which involves the velocities ^i, ^2> • • • 9n- It follows, by the same argument as in Art. 205, that the condition for * secular ' stability, when there are dissipative forces affecting the co-ordinates ?i> ?2> • • • ?n> but not the ignored co-ordinates x, x'> • • • > i® *^^ V + K should be a minimum. In the application to the problem of stationary waves, it will tend to clearness if we eliminate all infinities from the question by imagining that the fluid circulates in a ring-shaped canal of uniform rectangular section (the * "Die Energie der Wogen und des Windes," Berl Monaisber, July 17, 1890 [Wisa, Abh. t. iii. p. 333]. 254] Dynamical Condition for Permanent Type 421 sides being horizontal and vertical), of very large radius. The generalized velocity x corresponding to the ignored co-ordinate may be taken to be the flux per unit breadth of the channel, and the constant momentum of the circulation may be replaced by the cyclic constant k. The co-ordinates ?i> ?2> • • • ?n of the general theory are now represented by the value of the surface-elevation (tjI) considered as a function of the longitudinal space- co-ordinate X, The corresponding components of extraneous force are repre- sented by arbitrary pressures applied to the surface. If I denote the whole length of the circuit, then considering unit breadth of the canal we have F ' where r^ is subject to the condition = \gp\ -qHx, (5) J / y^dx = 0. (6) If we could with the same ease obtain a general expression for the kinetic energy of the steady motion corresponding to any prescribed form of the surface, the condition in either of the forms above given would, by the usual processes of the Calculus of Variations, lead to a determination of the possible forms, if any, of stationary waves*. Practically, this is not feasible, except by methods of successive approxi- mation, but we may illustrate the question by reproducing, on the basis of the present theory, the results already obtained for 'long ' waves of infinitely small amplitude. If h be the depth of the canal, the velocity in any section when the surface is maintained at rest, with arbitrary elevation 17, is x/(* +'?)» where x is the flux. Hence, for the cyclic constant, «=x/V+,)-><fe=^|(l4jV&:) (7) approximately, where the term of the first order in 17 has been omitted, in virtue of (6). The kinetic energy, \pKXi may be expressed in terms of either x or #c. We thus obtain the forms * For some general considerations bearing on the problem of stationary waves on the common surface of two currents reference may be made to Helmkoltz' paper. This also contains, at the end, some speculations, based on calculations of energy and momentum, as to the length of the waves which would be excited in the first instance by a wind of given velocity. These appear to involve the assumption that the waves will necessarily be of permanent type, since it is only on some such hypothesis that we get a determinate value for the momentum of a train of waves of small amplitude. 422 Surface Waves [chap, ix The variable part of V -Tq is and that of F + jST is ^'{'-w)jy^ ^''^ It is obvious that these are both stationary for 17 =0; and that they will be stationary for any infinitely small values of »;, provided x*=gh^, or K^=ghl^, K we put x=^^» ^^ jc =c/, this condition gives c^-=gh, (12) in agreement with Art. 175. It appears, moreover, that 1; =0 makes V +K a maximum or a minimum eu^cording as c' is greater or less than gh. In other words, the plane form of the surface is secularly stable if, and only if, c<J{gh). It is to be remarked, however, that the dissipative forces here contemplated are of a special character, viz. they affect the vertical motion of the surface, but not (directly) the flow of the liquid. It is otherwise evident from Art. 176 that if pressures be applied to maintain any given constant form of the surface, then if c^>gh these pressures must be greatest over the elevations and least over the depres- sions. Hence if the pressures be removed, the inequalities of the surface will tend to increase. Wave-Propagation in Two Dimensions. 255. We may next consider some cases of wave-propagation in two horizontal dimensions x, y. The axis of z being drawn vertically upwards, we have, on the hypothesis of infinitely small motions, 2 = |-,. + ^(0. (1) where <f> satisfies V ^ = (2) The arbitrary function F {t) may be supposed merged in the value of d^jdt. If the origin be taken in the undisturbed surface, and if J denote the elevation at time t above this level, the pressure-condition to be satisfied at the surface is '-j[tL «'> and the kinematical surface-condition is dt cf. Art. 227. Hence, for z = 0, we must have '^-[lU <« 8/« ' ^ dz or, in the case of simple-hannonic motion, S + ?^ = 0. (5) <rV = i7|t' (^) if the time-factor be e» (»'+'). 254-255] Wave-Propagatio7i in Two Dimeiisiojis 423 The fluid being supposed to extend to infinity, horizontally and down- wards, we may briefly examine, in the first place, the effect of a local initial disturbance of the surface, in the case of symmetry about the origin. The typical solution for the case of initial rest is easily seen, on reference to Art. 100, to be provided as in Art. 228. . sinai fc« T /7 \ J = cos at Jq (kw), J ^^ = 9h (7) (8) To generalize this, subject to the condition of symmetry, we have recourse to the theorem r 00 /•« JO ^0 (9) JO of Art. 100 (12). Thus, corresponding to the initial conditions, i =/(«>). '^0 = 0, (10) /•GO 'a /• OO we have ^z=g\ 6** Jq (feo) icZA; | f{a)jQ{ka)ada,\ . J <y ^0 I J = I COS at Jq {Jew) kdk I f (a) Jq (ka) ada, Jo Jo ) ....(11) If the initial elevation be concentrated in the immediate neighbourhood of the origin, then, assuming f"V(a) ^ada =1, (12) J we have <f> = 2^/* ^^^ «** «^o (*tD) kdk (13) Expanding, and making use of (8), we get If we put z = —r cos 0, w = rsind, (15) /.« -^ we have I e** Jq (kw) dk = -, (16) by Art. 102 (9), and thence* j\^jQikw)kr^dk=[l-Jl==n\^^ (17) * HobBon, Proc. Lond. Math, Soc. t. xxv. pp. 72, 73. This formula may, however, be dinpensed with; see the first footnote on p. 374 ante. 424 Surface Waves [chap, ix where ^ = cos 6 (cf. Art. 85). Hence From this, the value of J is to be obtained by (3). It appears from Arts. 84, 85 that ^2n+i(0) = 0, P,n(0) = (-)- ^'2'4".!^.''2n^^ > ••••(^^^ whence ^~2«i»M2!nj 6! U/ 10! UJ •••[•'• (20) « It follows that any particular phase of the motion is associated with a particular value of gi^lm, and thence that the various phases travel radially outwards from the origin, each with a constant acceleration. No exact equivalent for (20), analogous to the formula (21) of Art. 238 which was obtained in the two-dimensional form of the problem, and accord ingly suitable for discussion in the case where gt^jw is large, has been dis- covered. An approximate value may however be obtained by Kelvin's method (Art. 241). Since J^ {z) is a fluctuating function which tends as z increases to have the same period 27r as sin z^ the elements of the integral in (13) will for the most part cancel one another with the exception of those for which t da/dk = ro, or Aro == gt^im (21) nearly. Now when Jew is large we have •^0 (^) = ( J^)* sin (kw + iTT), (22) approximately, by Art. 194 (15), and we may therefore replace (13) by <f> = -i-^, -T [ e^ cos {<yt - kw - in) dk (23) Comparing with (6) and (8) of Art. 241, and putting now 2 = 0, we find as the surface value of <f> 6o = i — ^ sin (at — Aro), (24) 27rw^V \td^<^ldk^\ where k and a are to be expressed in terms of w and t by means of (8) and (21). Note has here been taken of the fact that d^/dk^ is negative. Since at = (gkt^)^ = 2kw, t d^ajdk^ = - Ighk'^ = - 2ro«/^«, . . (25) we have <*« = -~ — sin ?^ - (26) ♦ This result was given by Cauchy and Poiason. 255] Effect of a Local Impulse 425 The surface elevation is then given by (3). Keeping, for consistency, only the most important term, we find «= -/^cosg, (27) which agrees with the result obtained, in other ways, by Cauchy and Poisson. It is not necessi^ry to dwell on the interpretation, which will be readily understood from what has been said in Art. 240 with respect to the two- dimensional case. The consequences were worked out in some detail by Poisson on the hypothesis of an initial paraboloidal depression. When the initial data are of impulse, the typical solution is p^ = cos aie^ Jq (Arm), 1 = sin (7^ Jq (km), 9P (28) which, being generalized, gives, for the initial conditions P<l>o='F(m), J = (29) the solution <f> = - I COS aie^ Jq {Jew) kdk j F (a) Jq {ka) ada, ^•'^ -'^ I ....(30) 1 TOO /-OO ' ^ ' ^ = 1 a sin o/ Jo (^) *^* 1 ^ (o) Jo (^) ada. In particular, for a concentrated impulse at the origin, such that f F (a) 27rada =^ 1, (31) J 1 f* we find <f> = 5 — I cos <rfe*» Jq (kw) kdk (32) ZnrpJ Since this may be written ^ = 2^I/I^«"'^»(*«')*'^^' •• (33) we find, performing 1/gp . d/dt on the results contained in (18) and (20), ^ 1 If, (ft) gt* 2\P, (ft) (g<')« 3 ! f 3 (/*) ) \ ^ 27rpl r* 2! f» "^ 4! r^ •••|'| 2iT/)tD» (.* 5! Vro/ ' 9! (toj •"}• ) Again, when ^gt'/m is large, we have, in place of (27), 2*w/)m* C=-::/^sing (35) 42« Surface Waves [chap, ix 256. We proceed to consider the effect of a local disturbance of pressure advancing with constant velocity over the surface. This will give us, at all events as to the main features, an explanation of the peculiar system of waves which is seen to accompany a ship moving through sufl&ciently deep water. A complete investigation, after the manner of Arts. 242, 243, would be somewhat difficult ; but the general characteristics can readily be made out with the help of preceding results, the procedure being similar to that of Art. 249. Let us suppose that we have a pressure-point moving with velocity c along the axis of x, in the negative direction, and that at the instant under consideration it has reached the point 0. The elevation J at any point P may be regarded as due to a series of infinitely small impulses applied at equal infinitely short intervals at points of the axis of x to the right of 0. Of the annular wave-systems thus successively generated, those only will combine to produce a sensible effect at P which had their origin in the neighbourhood of certain points (?, which are determined by the consideration that the phase at P is 'stationary ' for variations in the position of Q, Now if t is the time which the source of disturbance has taken to travel from Q to 0, the phase of the waves at P, originated at Q, is S + i'^' (1) where w = QP (Art. 255 (35)). Hence the condition for stationary phase is ^-Y <2) Since, in this differentiation, and P are regarded as fixed, we have isr = c cos By where 6 = OQP ; hence Og= c^ = 2© sec fl (3) It is further evident that the points in the immediate neighbourhood of P, for which the resultant phase is the same as at P, will lie in a line perpendicular to QP. A glance at the above figure then shews that a curve 256] Waves due to a Travelling Disturbance 427 of uniform phase is characterized by the property that the tangent bisects the interval between the origin and the foot of the normal. If p denote the perpendicular from the origin to the tangent, and 6 the angle which p makes with the axis of x, we have, by a known formula, whence 2y=-gcot5, y = a cos^ 0, (5) The forms of the curves defined by (5) are shewn in the annexed figure*, which is traced from the equations x = y cos fl — -j^^ sin = Ja (5 cos — cos 3&), y = psinO -\- -j^ cos fl = — Ja (sin -\- sin 30), (6) The phase-difference from one curve to the corresponding portion of the next is 2n, This implies a difference 27rc^lg in the parameter a. Since two curves of the above kind pass through any assigned point P within the boundaries of the wave-system, it is evident that there are tioo corresponding effective positions of Q in the foregoing discussion. These are determined by a very simple construction. If the line OP be bisected in C, * Cf. Sir W. Thomson, "On Ship Waves," Proe. Irust Mech. Eng. Aug. 3, 1887 [Popular Lectures, t. iii. p. 482], where a similar drawing is given. The investigation there referred to, based apparently on the theory of * group- velocity/ was not published. See also R. E. Froude "On Ship Resistance," Papers of the Greenock Phil Soc. Jan. 19, 1894. 428 Surface Waves [chap. IX and a circle be drawn on CP as diameter, meeting the axis of xmR-i^R^, the perpendiculars PQi, PQ2 to PJBi, PR2, respectively, will meet the axis in the required points Q^, Q^. For CRi is parallel to PQi and equal to i-PQi > tl^c perpendicular from on PRi produced is therefore equal to PQi . Similarly, the perpendicular from on PR2 produced is equal to PQ^, The points Q^ , Q^ coincide when OP makes an angle sin-^ J, or 19° 28', with the axis of symmetry. For greater inclinations of OP they are imaginary. It appears also from (6) that the values of re, y are stationary when sin' = ^ ; this gives a series of cusps lying on the straight lines 1 X 2V2 = ± tan 19° 28'. (7) It will be seen immediately that a change of phase takes place from one portion of a curve to the other at the cusps. To obtain an approximate estimate of the actual height of the waves, in the different parts of the system, we have recourse to t6e formula (35) of Art. 255. If Pq denote the total disturbing pressure, the elevation at P due to the annular wave-system started at a point Q to the right of may be written «S = - 8 V2.p^* • «- £ • ^o«'' (8) where tn = pg, t = OQ This is to be integrated with respect to t, but (as already explained) the only parts of the integral which contribute appreciably to the final result will be those for which t has very nearly the values (tj, tj) corresponding to the special points Q^, Q2 &bove mentioned. As regards the phase, we have, writing t = r + t\ 9^ 4m '[tMiov&Aum^ <»' where, in the terms in [ ], i is to be put equal to r^ or tj as the case may be. 256] Ship- Waves 429 The secoad term vanishes by hypothesis, since the phase at P for waves started near Q^ or Q, ^ * stationary.' Again, we find Since «• = c cos d, w = , (10) w this gives, with the help of (2), [s>(£)]=l<»--°-« '"' Owing to the fluctuations of the trigonometrical term no great error will be committed if we neglect the variation of the first factor in (8), or if, further, we take the limits of integration with respect to t' to be ± oo . We have then, approximately, j-8^./:.-'°(s:+».^")*- where V = g^ (i - tan« dj), m,« = g^- (tan« », - i), (13) and the suffixes refer to the points Qi, Q, of the last figure. Since f cos mH'*dt' = [ sin mH'*dt' = V(if^)lm, (14) where the positive value of m is understood, we find i= (15) The two terms give the parts due to the transverse and lateral waves respectively. Since tOi = PQi = ^ct^ cos &i , tn, = PQ2 ==^0x2 cos flg? it appears that if we consider either term by itself, the phase is constant along the corre- sponding part of the curve p = xo = a cos* By whilst the elevation varies as \/2g*fo sec»g ^Vc»a*V|l-38in*e| ^'^^ 430 Surface Waves [chap, ix At the cusps, where the two systems combine, there is a phase-difference of a quarter- period between them*. The formulae make f infinite at a cusp, where sin* 5 = ^, but this is merely an indication of the failure of our approximation. That the elevation at a point P in the neighbourhood of a cusp would be relatively great might have been foreseen, since, as appears from (9) and (11), the range of points on the axis of x which have sent waves to P in sensibly the same phase is then abnormally extended. The infinity which occurs when 6 = \'tt is of a somewhat different character, being due to the artificial nature of the assumption we have made, of a pressure concentrated at a jxyint. With a diffused pressure this difficulty would disappear. It is to be noticed, moreover, that the whole of this investigation applies only to points for which gt^jAm is large; cf. Arts. 240, 255. It will be found on examination that this restriction is equivalent to an assumption that the parameter a is large compared with 27rc^/g. The argument there- fore does not apply without reserve to the parts of the wave-pattern near the origin. Although the mode of disturbance is different, the action of the bows of a ship may be roughly compared to that of a pressure-point of the kind we have been considering. The figure on p. 427 accounts clearly for the two systems of transverse and lateral waves which are in fact observed t, and for the especially conspicuous * echelon ' waves at the cusps, where these two systems coalesce. These are well shewn in the annexed drawing J by Mr R. E. Froude of the waves produced by a model. A similar system of waves is generated at the stern of the ship, which may roughly be regarded as a negative pressure-point. With varying speeds of the ship the stern-waves may tend partially to annul, or to reinforce, * The investigation in the preceding edition was defective through an oversight, but was corrected in the German translation, Leipzig, 1907. Kelvin returned to the subject in 1905: "Deep Sea Ship- Waves," Trans. E. 8. Edin. t. xxv. p. 1060 [Papers, t. iv. p. 407]. The distri- bution of surface elevation ai rived at differs however from that found in the text, owing to the adoption of a special law of pressure-intensity which is unfortunately not an adequate repre- sentation of a localized disturbance. See also Havelock, "The Propagation of Groups of Waves in Dispersive Media. . .," Proc. Roy, Soc. A, t. Ixzxi. p. 398 (1908). The waves due to a local obstacle at the bottom of a stream were investigated by Ekman, "On Stationary Waves in Running Water," Arkiv for Matem, t. iii. (1906). f A diagram of the forms of the wave-ridges, in which account is taken of the change of phase at the cusps, is given by Ekman, l.c. The whole system is best seen when viewed almost vertically from a great height on the precipitous .sides of a lake (such as Garda). If the water be quite smooth except for the passage of a steamer, the wave-pattern is seen beautifully developed in accordance with the diagram. X Copied, by permission of Mr Froude and the Council of the Institute of Naval Architects, from a paper by the late W. Froude, "On the Effect on the Wave-Making Resistance of Ships of Length of Parallel Middle Body," Trans. Inst. Nav. Arch. t. xvii. (1877). 256] Ship- Waves 431 the effect of the bow-waves, and consequently the wave-resbtance to the ship as a whole for a given speed may fluctuate up and down as the length of the ship is increased*. Cf. Art. 24i. To examine the modification produced in the wave-pattern when the depth of the water has to be taketi into account, the preceding argiunent must be put in a more general form. If, as before, ( is the time the pressure-point has taken to travel from Q to 0, it may be shewn that the phase of the disturbance at P, due to the impulse delivered at Q, will differ only by a constant from k{Vt-m), (17) if 'i/njk be the predominant wave-length in the neighbourhood of P, and F the corresponding wa ve- velocity f. This predominant wave-length is determined by the condition that the phase is stationary for variations of the wave-length only, i.e. ^.*(F(-nj) = 0, or x?=Vl, (18) where V, = d (kVydk, is the group- velocity (Art. 236). For the effective part of the disturbance at P, the phase (17) must further be stationary as regards variations in the position of Q; hence, differentiating partially with respect to (, we have w=V, or V = cco69 (19) • See W. Froude, I.c. and R, E. Proude, "On the Leading Phenomena of the Wave-Making Resistance of Ships," Traru. Intl. Nav. Arch. t. iiii. (1S61), where drawings of actual wave- patterns under varied conditions of speed are given, which are, as to their main features, in atriliing agreement witb the results of the above theory. Some of these drawings are reproduced in Kelvin's paper in the Proc. Irul. Mteh. Eng. above cited. For a discussion of the wave- resistance encountered by an ideal form of ship see Michell, Fha. Mag. (6), t. ilv. p. 106 (1898). t The symbol e,, which was previously employed in this seiwe, now denotes the velocity of tbe pressure-point over the water. 432 Surface Waves [chap, ix since m^ c cos 0. Now, referring to the figure on p. 426, we have p = ce cos fl - to = F< - n; (20) Hence for a given wave-ridge p will bear a constant ratio to the wave- length A, and in passing from one wave-ridge to the next this ratio will increase (or decrease) by unity. Sinq^ A is determined as a function of by (19), this gives the relation between p and 0. Thus in the case of infinite depth, the formula (19) gives c«cosa0= F« = ^, (21) and the required relation is of the form p = a cos« 0, (22) as above. When the depth (A) is finite, we have and the relation is c^cos^fl = F« = f^ tanh ^, (23) ?tanh - = 4 cos^e, (24) a p gfi where the values of a for successive wave-ridges are in arithmetic pro- gression. The forms of the curves in various cases have been sketched by Ekman*. Since the expression on the left-hand side cannot exceed unity, it appears that if c* > gh there will be an inferior limit to the value of d, determined by cos«fl = ^ (25) It follows that when the speed of the disturbing influence exceeds '\/{gh) the transverse waves disappear, and we have only the lateral waves. This tends to diminish the wave-making resistance (cf. Art. 249) f. • Standing Waves in Limited Masses of Water. 257. The problem of free oscillations in two horizontal dimensions (x, y), in the case where the depth is uniform and the fluid is bounded laterally by vertical walls, can be reduced to the same analytical form as in Art. 190. If the origin be taken in the undisturbed surface, and if ^ denote the elevation at time t above this level, the conditions to be satisfied at the free surface are as in Art. 255 (3), (4). * l.c. ante p. 430 t It is found that the power required to propel a torpedo-boat in relatively shallow water increases with the speed up to a certain critical velocity, dependent on the depth, then decreases, and finally increases again. See papers by Rasmussen, Trans. InsL Nav. Arch. t. xli. p. 12 (1899); Rota, ibid. t. xlii. p. 239 (1900); Yanow and Marriner, ibid. t. xlvii. pp. 339, 344 (1906). 256-257] Waves in Limited Masses of Water 433 The equation of continuity, V^ == 0, and the condition of zero vertical motion at the depth 2; = — A, are both satisfied by <f> = <f>i cosh A (2 4- A), (1) where <f>i is a function of x, y, such that '^' + 'I? + *V. - ..(2) The form of <f>i and the admissible values of k are determined by this equation, and by the condition that , ' ^-0 m at the vertical walls. The corresponding values of the * speed' (a) of the oscillations are then given by the surface-condition (6), of Art. 255; viz. we have a* Bs gk tanh kh (4) ik This makes ? = — sinh kh .<f>i (5) The conditions (2) and (3) are of the same form as in the case of small depth, and we could therefore at once write down the results for a rectangular or a circular* tank. The values of k, and the forms of the free surface, in the various fundamental modes, are the same as in Arts. 190, 191 1, but the amplitude of the osclQation now diminishes with increasing depth below the surface, according to the law (1) ; whilst the speed of any particular mode is given by (4). When kh is small, we have a* = k^gh, as in the Arts, referred to. We may also notice in this connection the case of a ]ong and narrow rectangular tank having near its centre one or more cylindrical obstacles, whose generating lines are vertical. The origin being taken at the centre of the free surface, and the axis of x parallel to the length /, we imagine two planes x = ±z* tohe drawn, such that x' is moderately large compared with the horizontal dimensions of the obstacles, whilst still small in comparison with the length (/). Beyond these planes we shall have S«0, car* + k^(t>i =0 (6) * For references to the original investigations by Poisson and Rayleigh of waves in a circular tank see p. 281. The problem was also treated by Merian, Ucber die Bewegung iropfbarer FlUasigkeilen in Offween, Basel, 1828 [see VonderMuhll, Maih. Ann. t. xxvii. p. 575] and by Ostrogradsky, "M^moire sur la propagation des ondes dans un bassin cylindrique," M4m. des Sav. JStrang, t. iii. (1832). t It may be remarked that either of the two modes figured on p. 280 may be easily excited by properly-timed horizontal agitation of a tumbler containing water. L. H. 28 434 Surface Waves [chap, ix approximately, and therefore, for x>z', (Pi^Asiakx+BcoBlcXt (7) whilst, for x< -x\ <l>i=iABmkx-B cos kx, (8) since in the gravest mode, which is alone here considered, 4> must be an odd function of x. In the region betiveen the planes x = ±:x' the configuration of the lines 0^= const, is, for a reason to be explained in Art. 290 in connection with other questions, sensibly the same as if in (2) we were to put k=0. So far as this region is concerned, the problem is in fact the same as that of conduction of electricity along a bar of metal which has the same form as the actual mass of water, and has accordingly one or more cylindrical perforations occupying the place of the obstacles. The electrical resistance between the two planes is then equivalent to that of a certain length 2x^ + a of an unperiorated bar of the same section. The difference of potential between the phases may be taken to be 2(kAx' -{-B), by (7), since kx' is small; and the current per unit sectional area is kA, approximately. Thus 2 {kAx' +B)=^(2x'+a)kA, (9) whence B/A =ika (10) and (t)i=A(Bmkx+ikacoskx), (11) for x>x\ The condition 80/3a? =0, to be satisfied for x=il, gives cos ikl- ika sin H (12) or, since ka\s& small quantity, cos iifc (^ + a) =0 .(13) The introduction of the obstacles has therefore the effect of virtually increasing the length of the tank by a. The period of the gravest mode is accordingly v=V(7-*-*^^') '''' wherer=Z+a. The value of a is known for one or two cases. In the case of a circular column of radius b, in the centre of the tank, the formulae (11) and (13) of Art. 64 shew that 0i varies as x+C, or x+nb^/a, practically, when x is comparable with the breadth a of the tank. Comparing with (11) above we see that a=2nb*/a, (16) subject to the condition that the ratio b/a must not exceed about J*. When the plane x=0 is occupied by a thin rigid diaphragm of breadth a, having a central vertical slit of breadth c, the formula is a=-logsec-^ (16) * The formula (14) was in this case found to be in good agreement with experiment (Lamb and Cooke, PhU. Mag. (6), t. zx. p. 303 (1910)). The experiments were made chiefly with a view to test the above method of approximation, which has other more important applications ; see Arts. 306, 307. 267-258] Waves in Limited Masses of Water 435 258. The number of cases of motion with a variable depth, of which the solution has been obtained, is very small. 1^. We may notice, first, the two-dimensional oscillations of water across a channel whose section consists of two straight lines inclined at 45^ to the vertical*. The axes of ^, z being respectively horizontal and vertical, in the plane of a cross- section, we assume <^+*^=il {cosh A;{y+t2)+oosifc(y+»2)}, (1) the time-factor cos (ai+t) being understood. This gives <f>=A(QOBhhyQO&kz+<iORhyGo&\ikz)y '^=^4 (sinhJEr^sinJb-sinibysinhib;). ..(2) The latter formula shews at once that the lines y = ±.z constitute the stream-line '^=0, and may therefore be taken as fixed boundaries. The condition to be satisfied at the free surface is, as in Art. 227, <^'<^=?^| (3) Substituting from (2) we find, if h denote the. height of the siuiace above the origin, a^ (cosh ky cos kh + cos ky cosh kh) =gk(- cosh kysinhk-k- cos ky sinh M). This will be satisfied for all values of y, provided o*' cos kh= -gk sin kh, o-' cosh kh =gk sinh kh, (4) whence tanh kh= - tan kh .....' (5) This determines the admissible values of k; the corresponding values of o* are then given by either of the equations (4). Since (2) makes an even function of y, the oscillations which it represents are sym- metrical with respect to the medial plane y =0. The asymmetrical oscillations are given by +ti^ =%A {cosh k {y+iz) - cos k (y +t2)}, (6) or <l}= -A{smhkyBmkz+ankyamhkz), ylr=A{coehkycoekz-oo8kyGoshkz). (7) The stream-line ifr =0 consists, as before, of the lines y = ^z; and the surface-condition (3) gives 0-' (sinh kyamkh+ sin ky sinh kh) =gk (sinh ky cos kh + sinky cosh kh). This requires a* sin kh =gk cos kh, a^ sinh kh =gk cosh kh, (8) whence tanh kh = tan kh (9) The equations (5) and (9) present themselves in the theory of the lateral vibrations of a bar free at both ends ; viz. they are both included in the equation cos m cosh m = 1, (10)t where m=2kh. The root kh =0, of (9), which is extraneous in the theory referred to, is now important ; it corresponds in fact to the slowest mode of oscillation in the present problem. Putting * Kirohhoff, **Ueb6r stehende Sohwingungen einer schweren Flussigkeit,'* Berl. Monatther. May 15, 1879 [Ges, Abh. p. 428]; Greenhill, Ic, ante p. 363. t Cf. Rayleigh, Theory of Sound, t. i. Art 170, where the numerical solution of the equation is fully disoussed. 28—2 436 Surface Waves [OHAP. IX Al^ -Bj and making k infinitesimal, the formulae (7) become, on lestoring the time-factor, and taking the real parts, =- 25^8 . cos (rr* + c), i/r=fi{y'-2;*).oos(a'<+€), (11) whilst from (8) The corresponding form of the free surface is (12) f=-r^1_ =2o-fi%.sin(c7<+€) (13) The surface in this mode is therefore always plane. The annexed figure shews the lines of motion (^ = const.) for a series of equidistant values of ifr. The next gravest mode is symmetrical, and is given by the lowest finite root of (5), which is ^^ = 2*3650, whence o- = 1*6244 {glhy. The profile of the surface has now two nodes, whose positions are determined by putting =0, z =^, in (2) ; whence it is found that I = ±-5516*. The next mode corresponds to the lowest finite root of (9), and so onf. 2^. Qreenhill, in the paper already cited, has investigated the symmetrical oscillations of the water across a channel whose section consists of two straight lines inclined at 60° to the vertical. In the (analytically) simplest mode of this kind we have, omitting the time- factor, +i^ =iA (y+izf +B, (14) or <^=ilz(2«-3y«)+fi, Vr=ily (y« -3z«), (16) the latter formula making i/r = along the boundary y = ± tJS.z. The surface-condition (3) is satisfied for z=h, provided <r^=glK B=2Ah^ (16) * Rayleigh, Theory of Sound, Art. 178. t An experimental verification of the frequencies, and of the positions of the loops (places of maximum vertical amplitude), in various fundamental modes, was made by Kirchhoff and Hansemann, "Ueber stehende Schwingungen des Wassers,'* Wied, Ann. t. x. (1880) [Kirchhoff, Oes, Ahh. p. 442]. 268-269] Transverse Oscillations in a Canal 437 The oorrespondin^ form of the free suifaoe, viz. f=J[tl-r-'^<**-'^)^'"<''^') ("> ia a parabolic cylinder, with two nodes at distances of '6774 of the half-breadth from the centre. The slowest mode, which must evidently be of asymmetrical type, has not yet been determined. 3°. If in any of the above cases we transfer the origin to either edge of the canal, and then make the breadth infinite, we get a system of standing waves on a sea bounded by a sloping bank. This may be regarded as made up of an incident and a reflected system. The reflection is complete, but there is in general a change of phase. When the inclination of the bank is 46° the solution is 0=JJ{e*»(oosifcy-ainA:y)+e~*''(cosAa+3inib)} C03{«r<+f) (18) For an inclination of 30° to the horizontal we have = JI {«*• sin ifcy + c-** <^'>'+*' sin \h (y - V3z) - V3e-**<^'>'-*> cos \k (y +V32)} cos (at +0 (19) In each case o-' =^glc, as in the case of waves on an unlimited sheet of deep water. These results, which may easily be verified ab initio, were given by Kirchhoff (Lc). 259. An interesting problem which presents itself in this connection is that of the transversal oscillations jof water contained in a canal of circuUiT section. This has not yet been solved, but it may be worth while to point out that an approximate determination of the frequency of the slowest mode, in the case where the free surface is at the level of the axis, can be effected by Rayleigh's method, explained near the end of Art. 168. If we assume as an 'approximate type' that in which the free surface remains always plane, making a small angle (say) with the horizontal, Jt appears, from Art. 72, 3<^, that the kinetic energy T is given by 22' = (t - I) />«*^'. (1) where a is the radius, whilst for the potential energy V we have 27 = |<7pa»e^ (2) If we assume that ac cos {at + c), this gives -^-ig^i <') whence a = 1*169 {gjay*. In the case of a rectangular section of breadth 2a, and depth a, the speed is given by Art. 257 (4), where we must put k = 7r/2a from Art. 178, and A == a. This gives a« = \tt tanh Jtt . ^ , (4) ♦ Rayleigh finds, as a closer approximation, <r = 1*1644 (gr/a)'; see PhU, Mag. (5), t. xlviii p. 666 (1899) [Papers, t. iv. p. 407]. 438 Surface Waves [chap, ix or a = 1*200 (g/o) • The frequency in the actual problem is less, since the kinetic energy due to a given motion of the surface is greater, whilst the potential energy for a given deformation is the same. Cf . Art. 45. 260. We may next consider the free oscillations of the water included between two transverse partitions in a uniform horizontal canal. Before proceeding to particular cases, we may examine for a moment the nature of the analytical problem. If the axis of x be parallel to the length, and the origin be taken in one of the ends, the velocity-potential in any one of the fundamental modes referred to may, by Fourier's Theorem, be supposed expressed in the form <f>= {Po + Pi cos kx-{- P^ cos 2hx + . . • + P, cos skx 4- . . . ) cos (a< + c), (1) where k = tt/J, if I denote the length of the compartment. The coefficients P, are here functions of y, z. If the axis of z be drawn vertically upwards, and that of y be therefore horizontal and transverse to the canal, the forms of these functions, and the admissible values of a, are to be determined from the equation of continuity VV = (^, (2) with the conditions that ^ ~ ^ (^) at the sides, and that a*<f> = 9 ^ (4:) at the free surface. Since d<f>ldx must vanish for a; = and x = l, it follows from known principles* that each term in (1) must satisfy the conditions (2), (3), (4) independently; viz. we must have gap gap 8^' + ll-' - '•*'^' = « (5) -*t W = « («) at the lateral boundary, and -'P' = 9^-W (^> at the free surface. The term Pq gives purely transverse oscillations such as have been discussed in Art. 258. Any other term Pg cos shx gives a series of fundamental modes with s nodal lines transverse to the canal, and 0, 1, 2, 3, ... nodal lines parallel to the length. - * See Stokes, "On the Critical Values of the Sums of Periodic Series,** Catnb. Trans, t. viii. (1847) [Papers, t. i. p. 236], 259-260] Waves in Uniform Caned 439 It will be sjifficient for our purpose to consider the term Pj cos kx. It is evident that the assumption ^ = Pj cos fee . cos ((rf -f €), (8) with a proper form of Pj and the corresponding value of a determined as above, gives the velocity-potential of a possible system of standing waves, of arbitrary wave-length 27r/i, in an unlimited canal of the given form of section. Now, as explained in Art. 229, by superposition of two properly adjusted systems of standing waves of this type we can build up a system of progressive waves <f> = P^C08{kxTat) (9) We infer that progressive waves of simple-harmonic profile, of any assigned wave-length, are possible in an infinitely long canal of any uniform section. We might go further, and assert the possibility of an infinite number of types, of any given wave-length, with wave-velocities ranging from a certain lowest value to infinity. The types, however, in which there are longitudinal nodes at a distance from the sides are from the present point of view of subordinate interest. Two extreme cases call for special notice, viz. where the wave-length is very great or very small compared with the dimensions of the transverse section. The most interesting types of the former class have no longitudinal nodes, and are covered by the general theory of 'long ' waves given in Arts. 169, 170. The only additional information we can look for is as to the shapes of the wave-ridges in the direction transverse to the canal. In the case of relatively short waves, the most important type is one in which the ridges extend across the canal with gradually varying height, and the wave-velocity is that of free waves on deep water as given by Art. 229 (6). There is another type of short waves which may present itself when the banks are inclined, and which we may distinguish by the name of 'edge- waves,' since the amplitude diminishes exponentially as the distance from the bank increases. In fact, if the amplitude at the edges be within the limits imposed by our approximations, it will become altogether insensible at a distance whose projection on the slope exceeds a wave-length. The wave- velocity is less than that of waves of the same length on deep water. It does not appear that the type of motion here referred to is very important. A general formula for these edge- waves has been given by Stokes*. Taking the origin in one edge, the axis of z vertically upwards, and that of y * "Report on Recent Researohes in Hydrodynamics," Brit. Ass. Rep. 1846 [Papers, t. i. p. 167]. 440 Surface Waves [chap, ix transverse to the canal, and treating the breadth as relative^ infinite, the formula in question is 4> = jye-»(»««^-*«^^' cos Jk (a; - cf), (10) where j3 is the slope of the bank to the horizontal, and c=(|8m/3)* (11) The reader will have no difficulty in verifying this result. 261. We proceed to. the consideration of some special cases. We shall treat the question as one of standing waves in an infinitely long canal, or in a compartment boimded by two transverse partitions whose distance apart is a multiple of half the arbitrary wave-length (27r/A;), but the investigations can easily be modified as above so as to apply to progressive waves, and we shall occasionally state results in terms of the wave- velocity. 1°. The solution for the case of a rectangular section, with horizontal bed and vertical sides, could be written down at once from the results of Arts. 190, 257. The nodal lines are transverse and longitudinal, except in the case of a coincidence in period between two distinct modes, when more complex forms are possible. This will happen, for instance, in the case of a square tank. o 2^. In the case of a canal whose section consists of two straight lines inclined at 45 to the vertical we have, first, a type discovered by Kelland* : viz. if the axis of x coincide with the bottom line of the canal, ky Icz (f) =A cosh -t| cosh j^ cos kx . cos ((ri +c) (1) This evidently satisfies V*0 =0, and makes |=4t' (2) for y = ±2, respectively. The surface-condition (Art. 260 (4)) then gives '^•=3*-^|' <') where k is the height of the free surface above the bottom line. If we put <r =fcc, the wave- velocity c is given by '="=72ifc*''"\^ <*> where k =27r/X, if X be the wave-length. When hfk is small, this reduces to c={\gh)K (5) in agreement with Art. 170 (13), since the mean depth is now denoted by \h. When, on the other hand, hjX is moderately large, we have c2__?_ (6) ft * " On Waves," Trans, R. 8. Edin, t. xiv. (1839). 260-261] Canal of Triangvlar Section 441 The formula (1) indioatoe now a rapid increase of amplitude towards the sides. We have here, in fact, an instance of * edge- waves,' and the wave-velocity agrees with that obtained by patting /9=45^ in Stokes' formula. The remaining types of oscillation which are symmetrical with respect to the medial plane y =0 are given by the formula <^ =0 (cosh ay cos jSz +cos fiy cosh az) cos lex . cos (o-^ +c) (7) provided a, /9, o- are properly determined. This evidently satisfies (2), and the equation of continuity gives a«-i9*=it* (8) . The surface-condition, Art. 260 (4), to be satisfied for z =%, requires a^ cosh ah=ga sinh aA, o-' cos fih= -g^mnffh (9) Hence aAtanhaA-i-/9^tan/3A:=0 (10) The values of a, /3 are determined by (8) and (10), and the corresponding values of a are then given by either of the equations (9). If, for a moment, we write aj=aA, y=fih, •• (11) the roots are given by the intersections of the curve a;tanhx+t/tan^=0, (12) whose general form can be easily traced, with the hyperbola x«-y«=ibW (13) There are an infinite number of real solutions, with values of fih lying in the second, fourth, sixth, . . . quadrants. These give respectively 2, 4, 6, ... longitudinal nodes of the free surface.' When hjX is moderately large, we have tanh<iA = l, nearly, and /3^ is (in the simplest mode of this class) a little greater than \ir. The two longitudinal nodes in this case approach very closely to the edges as X i^ diminished, whilst the wave- velocity becomes practically equal to that of waves of length X on deep water. As a numerical example, assuming ffh = \*\x ^n-, we find ah = 10-910, kh = 10-772, c = 10064 (1^ . The distance of either nodal line from the nearest edge is then -12^. We may next consider the asymmetrical modes. The solution of this type which is analogous to Kelland's was noticed by Greenhill (Z.c). It is (f) —A sinh -^sinh -t^ cos kx . cos (vt +€), (14) with ^*=^°^**^^ ^^^' When kh is small, this makes (r*=g/h, so that the * speed' is very great compared with that given by the theory of 'long' waves. The oscillation is in fact mainly transversal, with a very gradual variation of phase as we pass along the canal. The middle line of the surface is of course nodal. When kh is great, we get * edge- waves,' as in the case of Eelland's solution. The remaining asymmetrical oscillations are given by <^ =A (sinh ay sin fiz -Hsin /3y sinh az) oob kz , ooe {ai -{-t) (16) This leads in the same manner as before to a«-i9*=*« (17) 442 SurfoM Waves [chap. IX and o-' sinh ak^ga cosh oA, cr' sin ^=gff oos fih, (18) whence oAcoth ah=&hcotfih (19) There are an infinite number of solutions, with values of /3A in the third, fifth, seventh, . . . quadrants, giving 3, 5, 7, ... longitudinal nodes, one of which is central. 3<*. The case of a canal with plane sides inclined at 60° to the vertical has been treated by Macdonald*. He has discovered a very comprehensive type, which may be verified as follows. The assumption (f) =sP cos kx , cos {(rl +c). (20) where P=.4 cosh ib+fi sinh Jb+ cosh ^?^^C7 cosh ^+ 1) sinh ^^ (21) evidently satisfies the equation of continuity ; and it is easily shewn that it makes for y = ±V3«, provided C=2A, D= -2B The surface-condition, Art. 260 (4), is then satisfied, provided -f (A cosh kh +B sinh kk) =A sinh kk+B cosh kh, — r ( A cosh IT - -o smh gk\ 2 The former of these is equivalent to A =H ( cosh ^A - -T sinh kh and the latter then leads to (22) inh -^ j =A si . ,kh „ ,kh smh -^ - B cosh -5- . (23) y B=H(^coBhkh-smhkh\ (24) kh coth3 2-+l=0. \gk) gk Also, substituting from (22) and (24) in (21), we find P=H jcosh k{z-h)+^amhk(z - A)| (26) + 2H cosh ^^ jcosh k(^ + aV ^sinhifc^l + h\\. . . .(26) The equations (25) and (26) were arrived at by Maodonald, by a different process. The surface- value of P is Uh -« -( 1 + 2 oo8h M (coBb^-^ - ^ sinh ^)| . (27) The equation (25) is a quadratic in (r*/gk. In the case of a wave whose length (Zv/k) is great compared with h, we have nearly, and the roots of (25) are then ,, 3*A 2 *"**'' -2" =3iWi' 5=*^ and ^ = 1M (28) approximately. If we put a=kc, the former result gives c^ =igh, in accordance with the usual theory of Uong' waves (Arts. 169, 170). The formula (27) now makes P = 3H, approximately; •this is independent of y, so that the wave-ridges are nearly straight. The ♦ "Waves in Canals," Proc. Land. Math. Soc. t. xxv. p. 101 (1894). 261-262] Canal of Triangtdar Section 443 second of the roots (28) makes c' =g/h, giving a much greater wave- velocity ; but the con- siderations adduced above shew that there is nothing paradoxical in this. It will be found on examination that the cross-sections of the waves are parabolic in form, and that there are two nodal lines parallel to the length of the canal. The period is, in fact, almost exactly that of the symmetrical transverse oscillation discussed in Art. 258. When, on the other hand, the wave-length is short comp>ared with the transverse dimensions of the canal, kh is large, and coth §M = 1, nearly. The roots of (26) are then iJ='-<^p=4 (2«) approximately. The formef result makes P=H, nearly, so that the wave-ridges are straight, experiencing only a slight change of altitude towards the sides. The speed, o" = {gk)\ is exactly what we should expect from the general theory of waves on relatively deep water. If in this case we transfer the origin to one edge of the winter-surface, writing z+h for z, and y - JSh for y, and then make h infinite, we get the case of a system of waves travelling parallel to a shore which slopes downwards at an angle of 30"^ to the horizon. The result is <^ =H {e** +e-** <^*«'+" ,3g-4*(V8y-»)^ cos ib; . cos {at +e) (30) where c = (g/k)^. This admits of immediate verification. At a distance of a wave-length or so from the shore, the value of <^, near the suHace, reduces to • <f>=H^*coBkx.coB{at+€), (31) practically, in conformity with Art. 228. Near the edge the elevation changes sign, there being a longitudinal node for which ^ky=\og,2, (32) ory/X=127. The second of the two roots (29) gives a system of edge-waves, the results being equi- valent to those obtained by making /3=30° in Stokes* formula. OsdUations of a Spherical Mass of Liquid. 262. The theory of the gravitational oscillations of a mass of liquid about the spherical form is due to Kelvin*. Taking the origin at the centre, and denoting the radius vector at any point of the surface by a + f , where a is the radius in the imdisturbed state, we assume C'^^L, (1) 1 where 2|^ is a surface-harmonic of integral order n. The equation of con- tinuity V Y = is satisfied by OO Mil <f>-^^~Sn, (2) 1 »n * Sir W. Thomson, "Dynamical Problems regarding Elastic Spheroidal SbeUs and Spheroids of Incompressible Liquid,'* Phil Trana, 1863 [Papers, t. iii. p. 384]. 444 Surface Waves [oHAPt ix where /S^ is a surface-harmonic, and the kinematical coadition 3? _ ^ /o\ to be satisfied when r ^ a^ gives t— j»- • <«> The gravitation-potential at the free surface is (see Art. 200) 4ir7^»_£4iryP? (5) where y is the gravitation-constant. Putting g = ^7Tyf>a, r « a -f- Sf„, we find Q, = const. + fl'S -k — ti Sn (6) 1 Zii -f- 1 Substituting from (2) and (6) in the pressure-equation P = ^ _ £j + const., : (7) p 01 we find, since p must be constant over the surface, dSn 2 (n - 1) .. -ar=-2;rTr^^ <^^ Eliminating iS, between (4) and (8), we obtain dKn I 2n (n - 1) g y _ ,g. This shews that ^, oc cos (a„/. + e), where 2n (n — 1) g = 2n + 1 o* (10) For the same density of liquid, g ooa, and the frequency is therefore independent of the dimensions of the globe. The formula makes a^ = 0, as we should expect, since in the deformation expressed by a surface-harmonic of the first order the surface remains spherical, and the period is therefore infinitely long. ''For the case n « 2, or an ellipsoidal deformation, the length of the isochronous simple pendulum becomes fa, or one and a quarter times the earth's radius, for a homogeneous liquid globe of the same mass and diameter as the earth; and therefore for this case, or for any homogeneous liquid globe of about 5^ times the density of water, the half-period is 47 m. 12 s." 262-263] Oscillations of a Liquid Globe 445 "A steel globe of the same dimensionB, without mutual gravitation of its parts, could scarcely oscillate so rapidly, since the velocity of plane waves of distortion in steel is only about 10,140 feet per second, at which rate a space equal to the earth's diameter would not be travelled in less than Ih. 8m. 40s.*" When the surface oscillates in the form of a zonal harmonic spheroid of the second order, the equation of the lines of motion is xw*= const., where w denotes the distance of any point from the axis of symmetry, which is taken as axis of x (see Art. 95 (11)). The forms of these lines, for a series of equidistant values of the constant, are shewn in the figure. 263. This problem may also be treated very compactly by the method of 'normal co-ordinates ' (Art. 168). The kinetic energy is given by the formula -ip//^ dr dS, (11) where hS is an element of the surface r ^ a. Hence, when the surface oscillates in the form r^a-{-^^, we find, on substitution from (2) and (4), ^ - »?//^ ^dS. (12) * Sir W. Thomson, Lc. The exact theory of the vibrations of an elastic sphere gives, for the slowest oscillation of a steel globe of the dimensions of the earth, a period of 1 h. 18 m. See a paper "On the Vibrations of an Elastic Sphere," Proc Lond. Math, 8oc. t. xiii. p. 212 (1882). The vibrations of a sphere of incompressible substance, under the joint influence of gravity and elasticity, have been discussed by Bromwich, Proc. Lond. Math. Soc. t. xxx. p. 98 (1898). Th& influence of compressibility is examined by Love, Some Problems of Oeodynamica (Adams Prize Essay), Cambridge, 1911, p. 126. 446 Surface Waves [chap, ix To find the potential energy, we may suppose that the external surface is constrained to assume in succession the forms r = a-\' dt,^, where varies from to 1. At any stage of this process, the gravitation potential at the surface is, by (6), ii = const. + ?^^</fl^„ ...(13) Hence the work required to add a film of thickness ^,,80 is eS0.^-^^gpSSi„*dS (14) Integrating this from 6 = to 6 = 1, we find V == :^^gpSnn'dS (15) The results corresponding to the general deformation (1) are obtained by prefixing the sign S of summation with respect to n, in (12) and (15); since the terms involving products of surface-harmonics of different orders vanish, by Art. 87. The fact that the general expressions for T and V thus reduce to sums of squares shews that any spherical-harmonic deformation is of a 'normal type.' Also, assuming that f „ oc cos (cTnt + c), the consideration that the total energy T -\- V must be constant leads us again to the result (10). In the case of the forced oscillations due to a disturbing potential Q' cos (at + €) which satisfies the equation V*i2' = at all points of the fluid, we must suppose Q' to be expanded in a series of solid harmonics. If f„ be the equilibrium-elevation corresponding to the term of order n, we have, by Art. 168 (14), for the forced oscillation, ^" = i^:^^;V^*f»' .....(16) where -tj is the imposed speed, and a„ that of the free oscillations of the same type, as given by (10). The numerical results given above for the case n = 2 shew that, in a non- rotating liquid globe of the same dimensions and mean density as the earth, forced oscillations having the characters and periods of the actual lunar and solar tides would practically have the amplitudes assigned by the equilibrium- theory. 264. The investigation is easily extended to the case of an ocean of any uniform depth, covering a symmetrical spherical nucleus. Let b be the radius of the nucleus, a that of the external surface. The surface-form being r=a + j:*Cn (1) 263-264] Ocean of Uniform Depth 447 we aasiime, for the velocity-potential, « = {(^+l)5-„ + ^^i}^" (2) where the coefficients have been adjusted so as to make d<f)/dr=0 for r=b. The condition that ^ ~S^ '^' ^^' ,<„=ig.™ t- -«(-.! {©■-(ID I (« For the gravitation-potential at the free surface (1) we have "- . 3r ^i2n + l'"' ^^^ where po is the mean density of the whole mass. Hence, putting g =|9rypoa» we find ■ Q=oonBt.+ysr(i-2innp-,)f- <«> The pressure-condition at the free surface then gives The elimination of 8n between (4) and (7) leads to ^+(r»V.=0, (8) where «r.« = - iM—W^A 3 _pW _ If p =po, we have o-^ =0 as we should expect. When p>po the value of o-i is imaginary; the equilibrium configuration in which the external surface of the fluid is concentric with the nucleus is then imstable. (Cf. Art. 200.) If in (9) we put 6=0, we reproduce the result of the preceding Art. 'If, on the other hand, the depth of the ocean be small compared with the radius, we find, putting b=a-h, and neglecting the square of h/a. ...=n(n.l,(l-2-A.£)g (10) provided n be small compared with a/h. This agrees with Laplace's result, obtained in a more direct manner in Art. 200. But if n be comparable with a/h, we have, putting n =ka, so that (9) reduces to <r* -gk tanh kh, (II) as in Art. 228. Moreover, the expression (2) for the velocity-potential becomes, if we write r=a+z, =<l>i cosh A; (z +h)^ (12) where 0^ is a function of the co-ordinates in the surface, which may now be treated as plane. Cf. Art. 257. 448 Surface Waves [chap, ix The formulae for the kinetio and potential energies, in the genera] case, are easily found by the same method as in the preceding Art. to be and y=i9pj::{l-^,^)SiC,^d8. (14) The latter result shews, again, that the equilibrium configuration is one of minimum potential energy, and therefore thoroughly stable, provided p<po* In the case where the depth Ls relatively small, whilst n is finite, we obtain, putting b=a-h. ''=*T^2:,-(i^)j/f.«<i5. (15) whilst the expression for F is of course unaltered. If the amplitudes of the harmonics (n be regarded as generalized co-ordinates, the formula (15) shews that for relatively small depths the 'inertia-coefficients* vary inversely as the depth. We have had frequent illustrations of this principle in our discussions of tidal waves. CapiUarity. 265. The part played by Cohesion in certain cases of fluid motion has long been recognized in a general way, but it is only within comparatively recent years that the question has been subjected to exact mathematical treatment. We proceed to give some account of the remarkable investi- gations of Kelvin and Bayleigh in this field. It is beyond our province to discuss the physical theory of the matter*. It is sufficient, for our purpose, to know that the free surface of a liquid, or, more generally, the common surface of two fluids which do not mix, behaves as if it were in a state of uniform tension^ the stress between two adjacent portions of the surface, estimated at per imit length of the common boundary- line, depending only on the nature of the two fluids and on the temperature. We shall denote this 'surface-tension,' as it is called, by the symbol T^. The * dimensions' of Ti are MT"* on the absolute system of measurement. Its value in c.g.s. units (dynes per linear centimetre) appears to be about 74 for a water-air surface at 20° C.f ; it diminishes somewhat with rise of tem- perature. The corresponding value for a mercury-air surface is about 540. • For this, see Maxwell, Encyc. Briiann. Art. "Capillary Action" [Papers, Cambridge, 1890, t. ii. p. 641], where references to the older writers are given. Also, Rayleigh, *'0n the Theory of Surface Forces," PhU, Mag. (6), t. xxx. pp. 286, 466 (1890) [Papers, t. iii. p. 397]. t Rayleigh, "On the Tension of Water-Surfaces, Clean and Contaminated, investigated by the method of Ripples," Phil Mag. (6), t. zzz. p. 386 (1890) [Papers, t. iii. p. 394]; Pedersen, Pha. Trans. A, t. ccvii. p. 341 (1907); Bohr, PhU. Trans. A, t. ccix. p. 281 (1909). 264-266] Inflitence of Cohesion 449 An eqiuvalent statement is that the 'free' energy of any system, of which the surface in question forms part, contains a term proportional to the area of the surface, the amount of this 'superficial energy ' (as it is usually termed) per unit area being equal to Ti*. Since the condition of stable equilibrium is that the free energy should be a minimum, the surface tends to contract as much as is consistent with the other conditions of the problem. The chief modification which the consideration of surface-tension will introduce into our previous methods is contained in the theorem that the fluid pressure is now discontinuous at a surface of separation, viz. we have p-^'-'^^ihi)' where y, p' are the pressures close to the surface on the two sides, and Ri, J?, are the principal radii of curvature of the surface, to be reckoned negative when the corresponding centres of curvature lie on the side to which the accent refers. This formula is readily obtained by resolving along the normal the forces acting on a rectangular element of a superficial film, boimded by lines of curvature; but it seems unnecessary to give here the proof, which may be found in most modern treatises on Hydrostatics. 266. The simplest problem we can take, to begin with, is that of waves on a plane surface forming the common boundary of two fluids at rest. If the origin be taken in this plane, and the axis of y normal to it, the velocity-potentials corresponding to a simple-harmonic deformation of the common surface may be assumed to be <f> = Ce^^ cos kx . cos (ai + €), 1 .- . ^' = C'e^^^ cos kx . cos (at + c), j where the former equation relates to the side on which y is negative, and the latter to that on which y is positive. For these values satisfy V^ = 0, V^' = 0, and make the velocity zero f or y = T oo , respectively. The corresponding displacement of the surface in the direction of y will be of the type 7) = a COS kx . ain (ai -\- €); (2) and the conditions that drj d(f> d<f> dt^ dy" dy * for y = 0, give aa = - JtC = AC' (3) * The distinction between *free* and * intrinsic' energy depends on thermo-dynamical principles. In the case of changes made at constant temperature with free communication of heat, it is with the 'free* energy that we are concerned. L. H. 29 450 Surface Waves [chap, ix If, for the moment, we ignore gravity, the variable part of the pressure is given by - = ~- = — =— e*" cos kx . sm (at -j- €h ot k ^ = -^ = e-*vcos kx , sin (at + €). p dt k (4) To find the pressure-condition at the common surface, we may calculate the forces which act in the direction of y on a strip of breadth 8x, The fluid pressures on the two sides have a resultant (p' —p) Sx, and the difference of the tensions parallel to y on the two edges gives S (Tidrj/dx). We thus get the equation P-P' + T^l^» = 0. (5) to be satisfied when y = approximately. This might have been written down at once as a particular case of the general surface condition (Art. 265). Substituting in (5) from (2) and (4), we find ^« = /_^^,, (6) which determines the speed of the oscillations of wave-length 27r/i. The energy of motion, per wave-length, of the fluid included between two planes paraUel to xy, at unit distance apart, is ^■»'/:[*ii.*-»''/:[*'iL'- '" If we assume 17 =a cos kx « (8) where a depends on t only, and therefore, having regard to the kinematical conditions, = - k-^a^ cos kx, <!>' =k-^ de-*» cos fee, (9) we find T = J (p +p') ^-l a« . X (10) Again, the energy of extension of the surface of separation is -'■■/:^(i)r^-''.-»''-/:®"^ '••••'■■' Substituting from (8), this gives 7 =iTiitV . X (12) To find the mean energy, of either kind, per unit area of the common surface, we must omit the factor X. If we assume that a occos {at +c), where o- is determined by (6), we verify that the total energy T + V is constant. Conversely, if we assume that 17=2 (acosibx+^sinArx), (13) it is easily seen that the expressions for T and V will reduce to sums of squares of a, 4 and a, /9, respectively, with constant coefficients, so that the quantities a, fi are * normal co-ordinates.' The general theory of Art. 168 then leads independently to the formula (6) for the speed. By compounding two systems of standing waves, as in Ajrt. 229, we obtain a progressive wave-system 7y = a cos (fee =F at), (14) 266] Capillary Waves 451 travelling with the velocity "" " A "" V+ p') ' or, in terms of the wave-length, (15) c=fMi,y.A-i (16) The contrast with Art. 229 is noteworthy ; as the wave-length is diminished, the period diminishes in a more rapid ratio, so that the wave-velocity increases. * Since c varies as A ~ % the group-velocity is, by Art. 236 (3), ^ = "-^5A = ^''- (17) The verification of the relation between group- velocity and transmission of energy is of some interest. Taking fj=acosk(ct -x)j (18) we find that the total energy per unit area of the surface is l(p+p')*cV+JT,ifcV=i(pV)^'«*» (1») by ( 10), ( 12 ). The mean rate at which work is done by fluid pressure at a plane perpendicular to X is found by a calculation similar to that of Art. 237 to be i (p +p') ikc»a* , (20) The rate at which surface tension does work at such a plane is Ti ^ ^ = T'l IcHa^ sin* it (c< - x), the mean value of which is i7'iifc«ca«=i (p + p') itc»a* (21) If we add this to (20), and divide by the second member of (19), the quotient is f c, in agreement with (17). The fact that the group- velocity for capillary waves exceeds the wave- velocity helps to explain some interesting phenomena to be referred to later (Arts. 271, 272). For numerical illustration we may take the case of a free water-surface ; thus, putting /> = 1, p' = 0, Ti = 74, we have the following results, the units being the centimetre and second*. Wave-length Wave- velocity Frequency •50 •10 •05 30 68 96 61 680 1930 * Cf. Sir W. Thomson, Papers, t. iii. p. 620. The above theory gives the explanation of the crispations observed on the surface of water contained in a finger-bowl set into vibration by stroking the rim with a wetted finger. It is to be observed, however, that the frequency of the capillary waves in this experiment is double that of the vibrations of the bowl; see Rayleigh, "On Maintained Vibrations/' PhU. Mag. (5), t. xv. p. 229 (1883) {Papers, t. ii. p. 188; Theory of Sound, 2nd ed., c. xx.]. 29—2 452 Surface Waves [chap* ix ( -jT — flr J a COS hx . sin {eft + c), (1) 267. When gravity is to be taken into account, the common surface, in equilibrium, will of course be horizontal. Taking the positive direction of y upwards, the pressure at the disturbed surface will be given by ^ = -^ - gy ^ - (^ + gj a COB kx. sin {at + c), approximately. Substituting in Art. 266 (5), we find .•='^;,*+,^ ™ Putting <7 = ic, we find, for the velocity of a train of progressive waves, c2==^_-/>;f4. Jl^i^J-ii-Vf + riV (3) p-hp' k^ p + p' l-{-8\k^ J' ^ ' where we have written ^ = ., -^^-, = r (4) p p-- p In the particular cases of Tj = and flr = 0, respectively, we fall back on the results of Arts. 232, !<>, and 266. There are several points to be noticed with respect to the formula (3). In the first place, although, as the wave-length {^jk) diminishes from oo to 0, the speed {a) continually increases, the wave-velocity, after falling to a certain minimum, begins to increase again. This minimum value (c,„, say) is given by cj = } ^* . 2 ((^r )t (5) (6) and corresponds to a wave-length ^=^-^/(7> In terms of A^ and c^ the formula' (3) may be written 5=i(^T") (') 'm >'*m shewing that for any prescribed value of c, greater than c^, there are two admissible values (reciprocals) of A/A^j. For example, corresponding to -= 1-2 1-4 1-6 1-8 20 we have 2-476 3-646 4-917 6*322 7-873 -404 -274 -203 -158 127, *m * The theory of the minimum wave-velocity, together with most of the substance of Arts. 266, 267, was giveif by Sir W. Thomson, '*Hydrokinetic Solutions and Observations," Phil. Mag, (4), t. xlii. p. 374 (1871) [Baltimore Lectures, p. 598]; see also NcUure, t. v. p. 1 (1871). 267] Waves under Gravity a7id Capillarity 453 to which we add, for future reference, sin-i ^ = 56°26' 45°35' 38°41' 33°45' 30°. c For sufficiently large values of A the first term in the formula (3) for c* is large compared with the second; the force governing the motion of the waves being mainly that of gravity. On the other hand, when A is very small, the second term preponderates, and the motion is mainly governed by cohesion, as in Art. 266. As an indication of the actual magnitudes here in question, we may note that if A/A^ > 3, the influence of cohesion on the wave-velocity amounts only to about 5 per cent., whilst gravity becomes relatively ineffective to a like degree if A/A^ < J. It has been proposed by Kelvin to distinguish by the name of 'ripples' waves whose length is less than An^. The relative importance of gravity and cohesion, as depending on the value of X, may be traced to the form of the expression for the potential energy of a deformation of the type Tf=aoo8kx (8) The part of this energy due to the extension of the bounding surface is, per unit area, '-^ (9) whilst the part due to gravity is i9{p-p')a* (10) As X diminishes, the former becomes more and more important compared with the latter. For a water-surface, using the same data as before, with (7=981, we find from (5) and (6) Xm = l-73, c„=23-2, the units being the centimetre and the second. That is to say, roughly, the minimum wave- velocity is about nine inches per second, or *45 sea-miles per hour, with a wave- length of two-thirds of an inch. Combined with the numerical results already obtained, this gives, for c= 27-8 32-6 371 41-8 46-4 "I •' ,, , , , .3 6-3 8-6 10-9 13-6 the values X = ^ ,^^ .^^ .3^ .37 22 in centimetres and seconds. If we substitute from (7) in the general formula (Art. 236 (3)) for the group-velocity, we find ^=«-*?A-«(>-jJ!:;J!:D <") Hence the group-velocity is greater or less than the wave-velocity, according as A $ Ani. For sufficiently long waves the group-velocity is practically equal to Jc, whilst for very short waves it tends to the value f c*. * Cf. Rayleigh, U.cc. anU p. 372. 454 Surface Waves [chap. IX The relations between wave-length and wave-velocity are shewn graphically in the annexed figure, where the dotted curves refer to the cases where gravity and capillarity act separately, whilst the full curve exhibits the joint effect. As explained in Art. 236, the group-velocity is represented by the intercept made by the tangent on the axis of ordinates. Since two tangents can be drawn to the curve from any point on this axis (beyond a certain distance from 0), there are two values of the wave-length corresponding to any prescribed value of the ^rroup-velocity Z7. These two values of A coincide when JJ has a certain (minimum) value, indicated by the point where the tangent to the curve at the point of inflexion cuts Oc \ and it may be easily shewn that we then have ^=y^(3 + 2V3) = 2-542, where c^ is the minimum t^^avc-velocity as above. V = -7670 A further consequence of (2) is to be noted. We have hitherto tacitly supposed that the lower fluid is the denser (t.e. p>p\ as is indeed necessary for stability when Tj is neglected. The formula referred to shews, however, that there is stability even when p<p\ provided '<''yV))* '''\ t.«. provided X be less than the wave-length Xm of minimum velocity when the denser fluid 267-268] Group-Vdocity 455 is below. Hence in the case of water above and air below the maximum wave-length con- sistent with stability is 1*73 cm. If the fluids be included between two parallel vortical walls, this imposes a superior lin^it to the admissible wave-length, and we learn that there is stability (in the two-dimensional problem) provided the interval between the walls does not exceed -86 cm. We have here an explanation, in principle, of a familiar experiment in which water is retained by atmospheric pressure in an inverted tumbler, or other vessel, whose mouth is covered by a gauze with sufficiently fine meshes *. 268. We next consider the case of waves on a horizontal surface forming the common boundary of two parallel currents U, ZJ'f. If ,we apply the method of Art. 233, we find without difficiilty that the condition for a stationary wave-profile is now pV^ + p'U'^ = ^^{p-p') + kT„ (1) the last term being due to the altered form of the pressure-condition which has to be satisfied at the surface. This may be written 'P U + P'U' ]' ^l P__-P' + . *Zi_ _ PP' m _ uy (2) p + p' 1 k-p + p'.+ p + p' (p + /.')*^ '' "^' (' The relative velocity of the waves, which is superposed on the mean velocity of the currents (Art. 233), is ± c, provided c* = Co* - 7-^.-, {V-uy, (3) ■ (p + p)* where Cq denotes the wave-velocity in the absence of currents. The various inferences to be drawn from (3) are much as in the Art. cited, with the important qualification that, since Cq has now a minimum value, viz. the Cn, of Art. 267 (5), the equilibrium of the surface when plane is stable for disturbances of all wave-lengths so long as U^V^\<^-±^,c^, (4) where $ = p'/p. When the relative velocity of the two currents exceeds this value, c becomes imaginary for wave-lengths lying between certain limits. It is * The case where the fluids are contained in a cylindrical tube was solved by MaxweU, Encyc. Briiann, Art. "Capillary Action " [Papers, t. ii. p. 585], and compared with some experi- ments of Duprez. The agreement is better than might have been expected when we consider that the special condition to be satisfied at the line of contact of the surface with the wall of the tube has been left out of account. t Cf. Sir W. Thomson, Phil Mag. (4), t. xlii. p. 368 (1871) [BaUimore Lectures, p. 590]. 456 SarfoAie Waves [chap, ix evident that in the alternative method of Art. 234 the time-factor e*'* will now take the form e±**+*^*, where a=^ ., ^ .. {V-Vy-c^^^k, j8 = j^Jfc|C7-C^'|. ...(5) 8 The real part of the exponential indicates the possibility of a disturbance of continually increasing amplitude. For the case of air over water we have 8 — -00129, Cm =23*2 (c.s.), whence the maximum value of \U -'T]'\ consistent with stability is about 646 centimetres per second, or (roughly) 12-5 sea-miles per hour*. For slightly greater values the instability will manifest itself by the formation, in the first instance, of wavelets of about two-thirds of an inch in length, which will continually increase in amplitude until they transcend the limits implied in our approximation. 269. The waves due to a local impulse on the surface of still water may be investigated to a certain extent by Kelvin's method (Art. 241). Since ^ = — 9^/9y at the surface, we have <i = — I cos aie^^ cos hcilc. rt = I — cos kxkdh. ... (1) Hence to conform to (6) of Art. 241 we must put (f>(k) = ik/p(r (2) If in Art. 267 (2) we put ^' = and write, for shortness, Tjp^r (3) we have a^ = gk+ TB (4) Let us first suppose that capillarity alone is operative, so that <7« = rA!» (5) Since S=t2"***. S = i^'**"* («> wefind * = iy.^2. <^ = iA'2v^. ^W^^li ^"^^ The procedure of Art. 241 then gives 1 / 4a!' \ ''^.*prM'^l27r7^"*">' ^^^ The test-fraction (9) of Art. 241 is now comparable with T^tjo?, and the approximation therefore cannot claim great accuracy except as regards the earlier stages of the disturbance at any point. It appears also from (8) that * The wind-velocity at which the surface of water actually begins to be ruffled by the forma- tion of capillary waves, so as to lose the power of distinct reflection, is much less than this, and is determined by other causes. This question is considered later (Chapter xl). 268-270] Waves due to a Local ImpvUe 457 the wave-length and period at any point begin by being infinitesimal, and continually increase. These several circumstances are in contrast with what holds in the case of gravity waves (Art. 240). We have seen (Art. 267) that when gravity is taken into account there are two wave-lengths corresponding to any assigned value of the group- velocity V which exceeds the minimum Vq. The particular wave-lengths correspond- ing to given values of x and t may be found by the geometrical methods of Art. 241. Analytically, putting dajdh = U = xjt, they are determined by the real values of k satisfying the equation ((7 + 3rP)« = 4a« (gy = ^' (^i + ri») (9) The approximate expression for rj will accordingly consist of two terms of the type (8) of Art. 241, so that we have two systems of waves superposed. For X < U^, Kelvin's method indicates that the disturbance is unimportant*. When x/VQt is sufficiently large the real solutions of (9) are * = i$*. * = l^. (10) approximately, as if gravity and capillarity were respectively alone operative. The conditions for the validity of Kelvin's approximation in this case, viz. that gt^/x and afl/T't^ should both be large, are to some extent opposed, but admit of being reconciled if x and t are both sufficiently great. The wave- length must in each case be small compared with x. The effect of a travelling disturbance can be written down from the general formulae of Art. 248. If /c^, /c, be the two wave-lengths corresponding to the wave- velocity c, it appears from the figure on p. 454 that if /c^ < /c„ we shall have Vi <c, U^> c. The result will be v-^/-^', [*>o/ (11) , = i_M e*'^. [x < 0] If we put <f> (k) = iP/pa (12) this will be found to agree, as an approximation, with the result of the more complete investigation which follows. 270. We resume the investigation of the effect of a steady pressure- disturbance on the surface of a running stream, by the methods of Arts. 242, 243, including now the effect of capillary forces. This will give, in addition to the former results, the explanation (in principle) of the fringe of ripples which is seen in advance of a solid moving at a moderate spoked through still water, or on the up-stream side of any disturbance in a uniform current. * Rayleigh, Phil Mag. (6), t. xxi. p. 180 (1911). 458 Surface Waves [chap, ix Beginning with a simple-harmonic distribution of pressure, we assume - = - a; + j3e*«' sin fee, 5? = _ y ^_ ^ky cos fee, (1) the upper surface coinciding with the stream-line ^ = 0, whose equation is y = j8 cos fee, (2) approximately. At a point just beneath this surface we find, as in Art. 242 (8), for the variable part of the pressure, Po = Pp {(*c* — flr) cos fe» + /ic sin kx}, (3) where jli is the frictional coefficient. At an adjacent point just above the surface we must have K = ^0 + I'l ^ = j8/>{(fej2 -g- PTO cos fee -h jLtc sin kx), . .(4) where T' is written for T^p, This is equal to the real part of We infer that to the imposed pressure Po = C cos kx (5) will correspond the surface-form _ ^ {kc^ — g — k^T^) cos fer -- /AC sin kx .^ ^^"^ (kc^-g- k^T'Y + |LL«c« ^^ Let us first suppose that the velocity c of the stream exceeds the minimum wave-velocity (Cn,) investigated in Art. 267. We may then write kc^^g-^ k^r =^r{k^K^){K^-k), (7) where k^, k^ are the two values of k corresponding to the wave- velocity c on still water; in other words, 27r//ci, 27r//c2 are the lengths of the two systems of free waves which could maintain a stationary position in space, on the surface of the flowing stream. We will suppose that k^> k^. In terms of these quantities, the formula (6) may be written ^ C {k — Ki) (k2 — k) cos fer — jLt' sin kx ,^. ^y-T' (k - K^Y (/c, - kY -I- ii!^ ' ^^^ where /a' = fic/T\ This shews that if /a' be small the pressure is least over the crests, and greatest over the troughs of the waves when k is greater that K2 or less than /c^, whilst the reverse is the case when k is intermediate to /Ci, /cg. In the case of a progressive disturbance advancing over still water, these results are seen to be in accordance with Art. 168 (14). 270-271] Surface- Distiirhance of a Stream 459 271. From (8) we can infer as in Art. 243 the effect of a pressure of integral amount P concentrated on a line of the surface at the origin, viz. we find (k — /Ci) (/cj — k) cos kx — fjf sin kx y = ^T-J (k - Kx)* {Kt - A)» + /x'» The definite integral ia the real part of dk (9) / ^ '^^^ (10) (A; -*ci)(Ka -*)-♦>' The dissipation-coefficient fi' has been introduced solely for the purpose of making the problem determinate; we may therefore avail ourselves of the slight gain in simplicity obtained by supposing fi' to he infinitesimal. In this case the two roots of the denomi- nator in (10) are where v= — - — . The integral (10) is therefore equivalent to ._j_{r^'?^-.._r_f^.i (11) These integrals are of the forms discussed in Art. 243. Since k^>ki, v is positive, and it appears that when x is positive the former integral is equal to 27rt6*''' + r f^ dk (12) and the latter to I j dk (13) Jo ^^Kj On the other hand, when x is negative, the former reduces to r« p — ikx J,FT7/* ('*) and the latter to -2wfc*''«* + f ^— dk (16) Jo ^ + «:i We have here simplified the formulae by putting ir =0 ajier the transformations. If we now discard the imaginary parts of our expressions, we obtain the results which immediately follow. When jLt' is infinitesimal, the equation (9) gives, for x positive, ^'•y^-— "^ sin ^.x + i- (X), (16) and, for x negative, '^^ y = - -^^^ sin K,x + i- (x), (17) X #^2 ""■ tj^ 1- n/ X 1 ff COS fee,, ""cosfcr,,] ,-Q. where F{^)= i 7 dk-\ ,— — dk\ (18) This function F (x) can be expressed in terms of the known functions Ci ki^x. 460 Surface Waves [chap. IX Si Ki^x, Ci K^x, Si /cjX, by Art. 243 (30). The disturbance of level represented by it is very small for values of a;, whether positive or negative, which exceed, say, half the greater wave-length (27r/ici). Hence, beyond some such distance, the surface is covered on the down-stream side by a regular train of simple-harmonic waves of length 2'jt/ki, and on the up-stream side by a train of the shorter wave-length 27r//c2. It appears from the numerical results of Art. 267 that when the velocity c of the stream much exceeds the minimum wave-velocity (Cm) the former system of waves is governed mainly by gravity, and the latter by cohesion. It is worth notice that, in contrast with the case of Art. 243, the elevation is now finite when x = 0, viz. we have ttTi 1 Q y K2 - log K2 — Ki K (19) This follows easily from (16) and (18). The figure shews the transition between the two sets of waves, in the case of k^ = S/c^. The general explanation of the effects of an isolated pressure-disturbance advancing over still water is now modified by the fact that there are ttoo wave-lengths corresponding to the given velocity c. For one of these (the shorter) the group-velocity is greater, whilst for the other it is less, than c. We can thus understand why the waves, of shorter wave-length should be found ahead, and those of longer wave-length in the rear, of the disturbing pressure. • It will be noticed that the formulae (16), (17) make the height of the up-stream capillary waves the same as that of the down-stream gravity waves ; but this result will be greatly modified when the pressure is diffused over a band of sensible breadth, instead of being concentrated on a mathematical line. If, for example, the breadth of the band do not exceed one-fourth of the wave-length on the down-stream side, whilst it considerably exceeds the wave-length of the up-stream ripples, as may happen with a very moderate 271] Waves and Ripples 461 velocity, the different parts of the breadth will on the whole reinforce one another as regards their action on the down-stream side, whilst on the up- stream side we shall have 'interference,' with a comparatively small residual amplitude. This point may be illustrated by assumiDg that the integral surface-pressure P has the distribution ^' = 1^^ (20) which is more diffused, the greater the value of 6. The method of calculation will be understood from Art. 244. The result is that on the down-stream side and on the up-stream side op ''=-pr(K;^)*"'**'^"*-^"- <22) where the terms which are insensible at a distance of half a wave-length or so from the origin are omitted. The exponential factors shew the attenuation due to diffusion ; this is greater on the side of the capillary waves, since k^>k^ . When the velocity c of the stream is less than the minimum wave- velocity, the factors of are imaginary. There is now no indeterminateness caused by putting ^ = ab initio. The surface-form is given by y P [* cosib The integral might be transformed by the previous method, but it is evident a priori that its value tends rapidly, with increasing x, to zero, on account of the more and more rapid fluctuations in sign of cos kx. The disturbance of level is now confined to the neighbourhood of the origin. For a; = we find y = ^— , - fl + - sin-i --,) (24) Finally we have the critical case where c is exactly equal to the minimum wave-velocity, and therefore kj = k^. The first term in (16) or (17) is now infinite, whilst the remainder of the expression, when evaluated, is finite. To get an intelligible result in this case it is necessary to retain the frictional coefficient fi. If we put fi =2a7*, we have (ifc-K)*+t>' = {ifc-(ic-i-w-ior)} {k-iK-m+iw)} (25) so that the integral (10) may now be equated to ^-Pir L , ^^ ' x ^^- r u-r^ -^M (26) 4m [J k-{K-xn +tm) J o k -(k +XS -iw) J 462 Surface Waves [chap, ix The formulae of Art. 243 shew that when w is small the most important part of this expression, for points at a distance from the origin on either side, is V^- . 2 TT i> *'*. : ( 27 ) It appears that the surface-elevation is now given by TT . y = - -71 cos (/ex - Jtt) (28) The examination of the effect of inequalities in the bed of a stream, by the method of Art. 246, must be left to the reader. 272. The investigation by Rayleigh*, from which the foregoing differs principally in the manner of treating the definite integrals, was undertaken with a view to explaining more fully some phenomena described by Scott Russell t and Kelvin J. " When a small obstacle, such as a fishing line, is moved forward slowly through still water, or (which of course comes to the same thing) is held stationary in moving water, the surface is covered with a beautiful wave- pattern, fixed relatively to the obstacle. On the up-stream side the wave-length is short, and, as Thomson has shewn, the force governing the vibrations is principally cohesion. On the down-stream side the waves are longer, and are governed principally by gravity. Both sets of waves move with the same velocity relatively to the water; namely, that required in order that they may maintain a fixed position relatively to the obstacle. The same condition governs the velocity, and therefore the wave-length, of those parts of the pattern where the fronts are oblique to the direction of motion. If the angle between this direction and the normal to the wave- front be called d, the velocity of propagation of the waves must be equal to Vo cos d, where v^ represents the velocity of the water relatively to the fixed obstacle. "Thomson has shewn that, whatever the wave-length may be, the velocity of propagation of waves on the surface of water cannot be less than about 23 centimetres per second. The water must run somewhat faster than this in order that the wave-pattern may be formed. Even then the angle Q is subject to a limit defined by Vq cos Q = 23, and the curved wave-front has a corresponding asymptote. "The immersed portion of the obstacle disturbs the flow of the liquid independently of the deformation of the surface, and renders the problem in its original form one of great difficulty. We may however, without altering the essence of the matter, suppose that the disturbance is produced by the ♦ /.c. ariU p. 389. t "On Waves," Erii. Aaa. Rep. 1844. X I'C, ante p. 452. 271-272] Effect of a Travelling Disturbance 463 application to one point of the surface of a slightly abnormal pressure, such as might be produced by electrical attraction, or by the impact of a small jet of air. Indeed, either of these methods — the latter especially — gives very beautifid wave- patterns*." The character of the wave-pattern can be made out by the method explained near the end of Art. 256. If we take account of capillarity alone, the formida (19) of the Art. cited gives c* cos« » = F« = =^ , (1) by Art. 266, and the form of the wave-ridges is accordingly determined by the equation p = a sec2 d (2)t This leads to a; = a sec d (1 - 2 tan^ 6), y = 3a sec tan ^ (3) When gravity and capillarity are both regarded, we have, by Art. 267, c«co8»»=F» = ^ + ^' (4) Hence, if we put c„ = (4</r')*. 6 = 27r(^y, (5) ^"^*^« co^=H6 + aJ' (^) where cos a = Cj^jc (7) The relation between p and 6 is therefore of the form or cos* ^ _ n / p a cos* a\ .„. ^s^i^^Ucos^a"^ TP r ^^ 2 = cos* » ± V(co8* d - cos* a) (9) The four straight lines for which ^ = ± a are asymptotes of the curve thus determined. The values of Jtt — a for several values of the ratio cjc^ have been given in Art. 267. When the ratio cjc^ is at all considerable, a is nearly equal to Jtt, and the asymptotes make very acute angles with the axis of x. The upper figure on the following page gives the part of the curve which is relevant to the physical problem in the case of c = lOCmJ. The ratio of the wave-lengths of * Rayleigh, Lc, t Since V is now > V, it appears from Art. 266 (20) that the constant a must be negative. } The necessary calculations were made by Mr H. J. Woodall. The scale of the figure does not admit of the asymptotes being shewn distinct from the curve. 464 Sfarface Waves [chap. IX 272-273] Wave-Patterm 465 the 'waves ' and the 'ripples ' in the line of symmetry is then, of course, very great. The curve should be compared with that which forms the basis of the figure on p. 427. As the ratio cjc^ is diminished, the asymptotes open out, whilst the two cusps on ^thei: side of th^.axis approach one another, coincide, and finally disappear*. The wave-system has then a configuration of the kind shewn in the lower diagram, which is drawn for the case where the ratio of the wave- lengths in the line of symmetry is 4 : 1. This corresponds to a = 26° 34', or c=M2(;„t- When c<Cj^^ the wave-pattern disappears. 273. Another problem of great interest is the determination of the nature of the equilibrium of a cylindrical column of liquid, of circular section. This contains the theory of the well-known experiments of Bidone, Savart, and others, on the behaviour of a jet issuing under pressure from a small orifice in the wall of a containing vessel. It is obvious that the imiform velocity in the direction of the axis of the jet does not afiect the dynamics of the question, and may be disregarded in the analytical treatment. We will take first the two-dimensional vibrations of the column, the motion being supposed to be the same in each section. Using polar co-ordinates r, in the plane of a section, with the origin in the axis, we may write, in accordance with Art. 63, r* <f> = A — cos 80 . cos (of -i- €), (1) where a is the mean radius. The equation of the boundary at any instant will then be r = a+f, (2) sA where . . { .== coa 80 .^n {at + e)^ (3) the relation between the coefficients being determined by dt^ dr' • ^*^ for r == a. For the variable part of the pressure inside the column, close to the surface, we have ^ = -^ = — (7-4 cos «& . sin (o< -i- c) (5) p ot The curvature of a curve which differs infinitely little from a circle having its centre at the origin is found by elementary methods to be * A tentative diagram shewed that they were nearly ooincident for c=2cm (a =60^). t The figure may be compared with the drawing, from observation, given by Scott Russell, Lc, L.H. 30 466 Surface Waves [chap, ix or, in the notation of (2), n^a-a^^-^W) (®) Hence the surface-condition T p = -^ + const. (7) gives, on substitution from (5), cr« = «(««- 1) ^, (8)* For « = 1, we have a = 0; to our order of approximation the section remains circidar, being merely displaced, so that the equilibrium is neutral. For all other integral values of «, o^ is positive, so that the equilibrium is thoroughly stable for two-dimensional deformations. This is evident d priori, since the circle is the form of least perimeter, and therefore least energy, for given sectional area. In the case of a jet issuing from an orifice in the shape of an ellipse, an equilateral triangle, or a square, prominence is given to the disturbance of the type « = 2, 3, or 4, respectively. The motion being steady, the jet exhibits a system of stationary waves, whose length is equal to the velocity of the jet multiplied by the period {27t/g)'\. 274. Abandoning now the restriction to two dimensions, we assume that (f> = (f>i cos kz . cos (of + e), (9) where the axis of z coincides with that of the cylinder, and <f>i is a function of the remaining co-ordinates x, y. Substituting in the equation of continuity, V«^ = 0, we get (Vx> - *«) ^1 = 0, (10) where Vj* = 3"/9x* + 9V9y*' I^ w® P^^ a; = r cos d, y = r sin d, this may be written g^+^a, +r»g^ A^i-U (11) This equation is of the form considered in Arts. 101, 191, except for the sign of Jk* ; the solutions which are finite f or r = are therefore of the type ^1 = B/. (*r) ^'1 «0, (12) * For the original investigation, by the method of energy, see Rayleigh, "On the Instability of Jets," Proc, Lond. Math. Soc. t. z. p. 4 (1878), and "On the Capillary Phenomena of Jets," Proc, Boy. Soc. t. xxix. p. 71 (1879) [Papers, t. i. pp. 361, 377 ; Theory of Sound, 2nd ed. c. xx.]. The latter paper contains a comparison of the theory with experiment. f It is assumed that this wave-length is large compared with the circumference of the jet. Otherwise, the formula (18) must be employed, with <r = kc, where c is the velocity of the jet. 273-274J VibrcUions of a CylindriccU Jet 467 where, as in Art. 210 (11), • ^^^ " 2^7 ! r "^ 2 (2« + 2) "^ 2 . 4 (2«' + 2) (2« + 4) "^ • • T '"^^"^^ Hence, writing <f> — BIf (kr) cos 80 cos fa , cos (o< + c), (14) . we have, by (4), l=.-B ^^' ^^^ cos «» cos fa . sin (o< + €) (15) To find the sum of the principal curvatures, we remark that, as an obvious consequence of Euler's and Meunier's theorems on curvature of surfaces, the curvature of any section differing infinitely little from a principal normal section is, to the first order of small quantities, the same as that of the principal section itself. It is sufficient therefore in the present problem to calculate the curvatures of a transverse section of the cylinder, and of a section through the axis. These are the principal sections in the undisturbed state, and the principal sections of the deformed surface will make infinitely small angles with them. For the transverse section the formula (6) applies, whilst for the axial section the curvature is — 3*C/3«* ; so that the required sum of the principal curvatures is l + l-'^-Kr + ^S). = 1 _ gMjL^ (jfc«c« + »> - 1) COS sd cm hi. Bin (at + e) (16) a aa* Also, at the surface, ^ = ^ = — aBl, (ka) cos «d cos Xa; . sin ((rf + c) (17) p ot The surface-condition of Art. 265 then gives °*- w'*'°*-''-"-^' *'" For « > 0, cr' is positive ; but in the case {s = 0) of symmetry about the axis c^ will be negative if ha<\\ that is, the equilibrium is unstable for disturbances whose wave-length (^jk) exceeds the circumference of the jet. To ascertain the type of disturbance for which the instability is greatest, we require to know the value of ha which makes /o (ha) a maximum. For this Rayleigh finds i*a* = '4858, whence, for the wave- length of maximum instability, 2ir/it = 4-508 x 2a. 30—2 468 Sfwrface. Waves [ohap. ix There is a tendency therefore to the production of bead-like swellings and contractions, of this wave-length, with continually increasing amplitude, until finally the jet breaks up into detached drops'^. 275. This leads naturally to the discussion of the small oscillations of a drop of liquid about the spherical formf. We will slightly generalize the question by supposing that we have a sphere of liquid, of density p, surrounded by an infinite mass of other liquid of density p\ Taking the origin at the centre, let the shape of the common surface at any instant be given by r = a -i- i = a -i- iSn . sin (<rf + €), (1) where a is the mean radius, and S^ is a surface-harmonic of order n. The corresponding values of the velocity-potential will be, at internal points, ^ = - ^ ^ -S, . COS (o< + «), (2) and, at external points, since these make ^ ^ "" ^ ^ "" ^' for r =^ a. The variable parts of the internal and external pressures at the surface are then given by p = .*.". + ^- — Sn . sin (of + c), y' = . . . — ^"TT ^" . sin (o< -i- €). ... (4) To find the sum of the curvatures we make use of the theorem of SoUd Geometry, that if A, ft, v be the direction-cosines of the normal at (x, y, z) to that surface of the family jP (aj, y, z) = const. which passes through the point, viz. XV 1 . 1 dX dfjL dv ,^. ^^^"^ fii + ^ = ai + a^ + ai (s> * The argument here is that if we have a series of possible types of disturbance, with time* factors e'^ , e^ , e^ , . . ., where ai>a,>a3> . . ., and if these be excited simultaneously, the ampHtude of the first will increase relatiyely to those of the other components in the ratios ^(«i-«t) ^ ^ *!-«») ^ ^ ^ ^ ^ The component with the greatest a will therefore ultimately predominate. The instability of a cylindrical jet surrounded by other fluid has been discussed by Rayleigh, "On the Instability of Cylindrical Fluid Surfaces," PhU. Mag. {$), t. zxxiy. p. 177 (1892) [Papers, t. iii. p. 694]. For a jet of air in water the wave-length of maximum instability is found to be 6*48 X 2a. t Rayleigh, Ic. anU p. 466; Webb, Mess, of Math. t. ix. p. 177 (1880). 274-275] Vibrations of a Globule 469 Since the square of ([ is to be neglected, the equation (1) of the harmonic spheroid may also be written r-o + t., (6) where U = ^S^.an(at + e) (7) i.e. {n is A '0^ harmonic of degree n. We thus find r dy /*-!-^ + nS{..)- (8) whence 1,1 2 n{n + l)y 2 . (n - 1) (n + 2) _ • ._ , , ,„, Substituting from (4) and (9) in the general surface-condition of Art. 265, we find .' = n(n+l)(«-l)(n + 2)^^ ^^^^J^^^^^^. (10) If we put p' = 0, this gives «r» = n(n-l)(n + 2)^,. (11) The most important mode of vibration is that for which n = 2 ; we then have cr«=: pa^' Hence for a drop of water, putting jTi = 74, p — 1, we find, for the frequency, a/2ir = 3'87a"* vibrations per second,. if a be the radius in centimetres. The radius of the sphere which would vibrate seconds is a » 2'47 cm. or a little less than an inch. The case of a spherical bubble of air, surrounded by liquid, is obtained by putting p » in (10), viz. we have a«=(n+l)(n-l)(n+2)^3 (12) r For the same density of the liquid, the frequency of any given mode is greater than in the case represented by (11), on account of the diminished inertia : cf. Art. 91 (7), (8). «- • CHAPTER X WAVES OF EXPANSION 276. A TBEATISE on Hydrodynamics would hardly be complete without some reference to this subject, if merely for the reason that all actual fluids are more or less compressible, and that it is only when we recognize this compressibility that we escape such apparently paradoxical results as that of Art. 20, where a change of pressure was found to be propagated instofUaneously through a liquid mass. We shall accordingly investigate in this Chapter the general laws of propagation of small disturbances, passing over, however, for the most part, such details as belong more properly to the Theory of Sound. In most cases which we shall consider, the changes of pressure are small, and may be taken to be proportional to the changes in density, thus Ap = K . —, P where k (^ p dp/dp) is a certain coefficient, called the 'elasticity of volume.' For a given liquid the value of k varies with the temperature, and (very slightly) with the pressure. For water at 15® C, k = 2*045 x 10^® dynes per square centimetre. The case of gases will be considered presently. Plane Waves, 277. We take first the case of plane waves in a imiform medium. The motion being in one dimension (x), the dynamical equation is, in the absence of extraneous forces, dt dx"" pdx^ pdpdx' ^ ' whilst the equation of continuity. Art. 7 (5), reduces to t + hl^)-^ (2) If we put p = po(l + «), (3) where c* 276-278] Plane Waves 471 where p^ is the density in the undisturbed state, a may be called the 'con- densation' in the plane x. Substituting in (1) and (2), we find, on the supposition that the motion is infinitely small, di'^J^Fx' ^ ^ !'--& <»' as above. Eliminating « we have =f=rgi (8) The equation (7) is of the form treated in Art. 170, and the complete solution is u =f{ct - a;) + J (c« + x), (9) representing two systems of waves travelling with the constant velocity c, one in the positive and the other in the negative direction of x. It appears from (5) that the corresponding value of 8 is given by C8^f{ct-x)-F{ct'{-x) (10) For a single wave we have u = ± cs, (11) since one or other of the functions/, F is zero. The upper or the lower sign is to be taken according as the wave is travelling in the positive or the negative direction. It is easily shewn in this case that the approximations involved in (4) and (5) are valid provided u is everywhere small compared with c. There is an exact correspondence between the above approximate theory and that of * long ' gravity- waves on water. If we write Tf/h for 8, and gh for ic/po, the equations (4) and (5), above, become identical with Art. 169 (3), (6). 278. With the value of k given in Art. 276, we find for water at 15° C. c = 1430 metres per second. The number obtained directly by Colladon and Sturm* in their experiments on the lake of Geneva was 1437, at a temperature of 8*^ C. * Ann. de Chim. et de Phys. t, zxxvi ( 1S27). It may be mentioned that the velocity of sound in water contained in a tube ib liable to be appreciably diminished by the yielding of tiie wall. See Helmholtz, Fortschritte d. Physik, t. iv. p. 119 (184S) [Wiss. Abh. tip. 242]; Korteweg, Wied, Ann. t. v. p. 526 (1878); Lamb, Manch, Mem. t. xlii. No. 1 (1898). 472 Waves of Expansion [chap, x In the case of a gas, if we assume that the temperature is constant, the value of ic is determined by Boyle's Law J=f, (1) viz. « = Po> (2) -y© '»> so that " ' _ This is known as the * Newtonian' velocity of sound*. If we denote by H the height of a 'homogeneous atmosphere' of the gas, we have po = gpoB^ and therefore o = (9H)i, (4) which may be compared with the formula (13) of Art. 170 for the velocity of *long' gravity- waves in liquids. For air at 0*^ C. we have as corresponding values Po = 76 X 13-60 X 981, po = '00129, in absolute o.G.S. units ; whence c = 280 metres per second. This is considerably below the value found by direct observation. The reconciliation of theory and fact is due to Laplace f. When a gas is suddenly compressed, its temperature rises, so that the pressure is increased more than in proportion to the diminution of volume ; and a similar state- ment applies of course to the case of a sudden expansion. The formula (1) is appropriate only to the case where the expansions and rarefactions are so gradual that there is ample time for equalization of temperature by thermal conduction and radiation. In most cases of interest, the alternations of density are exceedingly rapid ; the flow of heat from one element to another has hardly set in before its direction is reversed, so that practically each element behaves as if it neither gained nor lost heat. On this view we have, in place of (1), the 'adiabatic' law j'=C^y. (5) Po ^Po^ where, as explained in books on Thermodynamics, y is the ratio of the two specific heats of the gas. This makes K^yPo, (6) and therefore c = ^/f— ) = ViygB) , (7) ♦ Principia, Lib. ii. Sect. viii. Prop. 48. t The usual referenoe is to a paper "Sur la vitesse du son dans Tair et dans l*eau," Ann, de Chim. et de Phy$, t. iii p. 238 (1816) [M4canique COesU, Livre 12»«, o. iii (1823)]. But Poisson in a memoir of date 1607 (see p. 479) refers to this explanation as having been already given by Laplace. 278-279] Vdoeity of Sound 473 If we put y » 1*402*, the former result is to be multiplied by 1*184, whence c s 3S2 metres per second, which agrees very closely with the best direct determinations. The oonfidenoe felt by phydoistB in ihe soundness of Laplace's view is so oomplete that it is now usual to apply the formula (7) in the inverse manner, and to infer the values of y for various gases and vapours from observation of wave-velooities in them. In strictness, a similar distinction should be made between the ^adiabatic' and * isothermal* coefficients of elasticity of a liquid or a solid, but practically the difierenoe is unimportant^ Thus in the case of water the ratio of the two volume-elasticities is calculated to be 10012t. The effects of thermal radiation and conduction on air-waves have been studied theoretically by Stokes ^ and Rayleighf. When the oscillations are too rapid for equaliza- tion of temperature, but not so rapid as to exclude communication of heat between adjacent elements, the waves diminish in amplitude as they advance, owing to the dissipation of energy which takes place in the thermal processes. The effect of conduction will be noticed, along with that of viscosity, in the next Chapter. According to the law of Charles and Dalton P^BpB, (8) where d is the absolute temperature, and £ is a constant depending on the nature of the gas. The velocity of sound will therefore vary as the square root of 0. For several of the more permanent gases, which have sensibly the same value of y, the formula (7) shews that the velocity varies inversely as the square root of the density, provided the relative densities be deter- mined under the same conditions of pressure and temperature. • 279. The theory of plane waves can also be treated very simply by the Lagrangian method (Arts. 13, 14). If ^ denote the displacement at time t of the particles whose undisturbed abscissa is x, the stratum of matter originally included between the planes x and a; + Sx is at the time t-^-ht bounded by the planes a? + f and ^ +^ + (l + g|) S^> so that the equation of continuity is '>(i+i) = '>o' (1) * The value found by the most recent direct experiments. t Everett, XJniU and Physical Constants, X "An Examination of the possible effect of the Radiation of Heat on the Propagation of ^onnd,'' Pha. Mag. (4), t. i. p. 306 (1861) [Papers, t. iiL p. 142]. I Theory of Sound, Art. 247. In a paper ** On the Cooling of Air by Radiation and Conduction, and on the Propagation of Sound/' PhU. Mag, (6), t. xlvii. p. 308 (1899) [Papers, t. iv. p. 376], Rayleigh concludes on experimented grounds that conduction is much more effective in this respect %h$n radiation. 474 Waves of Bxpansion [chap. X where po ^^ ^^^ densitj in the undisturbed state. Hence if s denote the 'condensation' (/> — PoVpo* we have dj ^"^ (2) « = — r3' ax The dynamical equation, obtained by considering the forces acting on unit area of the above stratum, is a«| dp '*«a<«~ dx' (3) These equations are exact, but in the case of small motions we may write p = Po + K8y (4) (5) and Substituting in (3) we find dx = c« dx^' (6) 'i where c" = k/pq. The solution of (6) is the same as in Arts. 170, 277. 280. The kinetic energy of a system of plane waves is given by T = \poiUuHxdydz (1) where u is the velocity at the point (a?, y, z) at time t. The calculation of the intrinsic energy requires a little care. The work done by unit mass in expanding through a small range, from the actual volume v to the standard volume v^, is given to the second order of small quantities by the expression as is obvious on inspection from Watt's diagram. Putting P = Vli + K8, Vq-V = 8Vq, (2) "" we have i (P + Po) (Vo - v) = Po (% - v) 4- i (2> - yo) {% - v) *= Po (vo -v) + ^ks^Vq (3) If we take the sum of the corresponding expressions for all the mass- elements of the system, the term y© (^o ~ ^) will disappear whenever the conditions are such that the total change of volume is zero. This being assumed, we have, for the work done by the gas contained in any given region, in passing from its actual state to the normal state, the expression W = ^Kfffs^dxdydz (4) V Vr 279-281] Energy of Somid-Waves 475 So far, no assumptioii has been made as to the precise manner in which the transition takes place ; this will affect the value of ic. It is only in the case of adiabatic expansion that the expression (4) can be identified with the intrinsic energy' in the strict sense of the term. When the expansion is isothermal^ the expression gives what is known in Thermodynamics as the * free energy.* In a progressive plane wave we have c8 = ±Uy and therefore T ^ W. The equality of the two kinds of energy, in this case, may also be inferred from the more general line of argument given in Art. 174. In the Theory of Sound special interest attaches, of course, to the case of simple-harmonic vibrations. If a be the amplitude of a progressive wave of period 27r/a, we may assume, in conformity with Art. 279 (6), i = a cos {kx — of + €), (6) where k = a/Cy and the wave-length is accordingly A «= 2ir/i. The formidae (1) and (4) then give, for the energy contained in a prismatic space of sectional area unity and length A (in the direction x), T-hW^ iPo<^«* A, (6) the same as the kinetic energy of the whole mass when animated with the maximum velocity oa. The rate of transmission of energy across unit area of a plane moving with the particles situate in it is © -^ = p(ra 8an(kx - ai + f) (7) The work done by the constant part of the pressure in a complete period is zero. For the variable part we have Ap = ks = - K^= Kka sin (kx - trt + t) (8) Substituting in (7), we find, for the mean rate of transmission of energy, \Kvka^ = \pQfT^a* X c (9) Hence the energy transmitted in any number of complete periods is exactly that cor- responding to the waves which pass the plane in the same time, as we should expect, since, c being independent of X, the group- velocity is identical with the wave- velocity (cf . Art. 237). Waves of Finite AmjUitude. 281. If ;? be a function of p only, the equations (1) and (3) of Art. 279 give, without approximation, dH_^d^ dH ... di* ~ p„* dp • dx* • ^^' 476 Waves of Uxpansion [chap, x On the 'isothenaal' hypothesis that ^ = f (2) this becomes 5^ »= — . ai,a (3) In the same way, the 'adiabatic' rdation w p - fp""" Po \P0' leads to ?L^ _ rPo gg* (K\ (•-e These exact equations (3) and (5) may be compared with the similar equation for *long' waves in a uniform canal, Art. 173 (3). It appears from (1) that the equation (6) of Art. 279 could be regarded as exact if the relation between p and p were such that p't'""*"' («) Hence plane waves of finite amplitude can be propagated without change of type if, and only if, i'-J'o = /»«c«(l-^) (7) A relation of this form does not hold for any known substance, whether at constant temperature or when free from gain or loss of heat by conduction and radiation*. Hence sound-waves of finite amplitude must inevitably undergo a change of type as they proceed. 282. The laws of propagation of waves of finite amplitude have been investigated, independently and by difEerent methods, by Eamshaw and Riemann. It is proposed to give here a brief account of these investigations, referring for further details to the original papers, and to the very full discussion of the matter by Rayleighf. Riemann's method:): has already been applied in this treatise to the discussion of the corresponding question in the theory of 'long' gravity- waves on liquids (Art. 187). He starts from the Eulerian equations (1) and (2) of Art. 277, which may be written 9« + „9«=_1^9p, (1) ot ox p dp ox di^'^Fx-'-^d-x <^^ * The relation would make p negative when p falls below a certain value. t Theory of Sound, 0. zi X **Ueber die Fortpflanzung ebener Loftwellen von endlicher Schwingungsweite/' Q6U, Ahk* t. viii. p. -43 riM&-9) {Wtfkt, 2** Aufl., Leipzig, 1892, p. 167]. 281-282] Waves of Finite Amplitude 477 Ifweput P=/(p) + t*, Q^f{p)-u, (3) where / (p) is a-s yet undetermined, we find, multipljdng (2) by /' (p), and adding to (1), dP dP_ ldpdp_ /// x3t* If we now determine / (p) so that </'(''»*=^.| <*> this may be written dP , dP -, , , dP ,^. 8? + ^^=-^-^(^)^ "(^^ In the aame way we obtain The condition (4) is satisfied by Substituting in (5) and (6), we find Hence dP «= 0, or P is constant, for a geometrical point moving with the velocity ^ S-(|)'+«. « whilst Q is constant for a point whose velocity is i-(^)*+» <'») Hence, any given value of P moves forward, and any value of Q move& backward, with the velocity given by (9) or (10), as the case may be. These results enable us to imderstand, in a general way, the nature of the motion in any given case. Thus if the initial disturbance be confined to the space between the two planes a; = a, a; » 6, we may suppose that P and Q both vanish for a; > a and for x<h. The region within which P is variable will advance, and that within which Q is variable will recede, until after a time these regions separate and leave between them a space for which P = 0, Q » 0, and in which the fluid is therefore at rest. The original 478 Waves of JEocpansion [chap, x disturbance lias thus been split up into two progressive waves travelling in opposite directions. In the advancing wave we have Q = 0, and therefore «=/0»), (11) so that both the density and the particle- velocity are propagated forwards at the rate given by (9). Whether we adopt the isothermal or the adiabatic law of expansion, this velocity of propagation will be found to be greater, the greater the value of p. The law of progress of the wave may be illustrated by drawing a curve with x as abscissa and p as ordinate, and making each point of this curve move forward with the appropriate velocity, as given by (9) and (11). Since those parts move faster which have the greater ordinates, the curve will eventually become at some point perpendicular to z. The quantities du/dx, dp/dx are then infinite ; and the preceding method fails to yield any information as to the subsequent course of the motion. Cf . Art. 187. 283. Similar results can be deduced from Eamshaw's investigation*, which is, however, somewhat less general in that it applies only to a pro- gressive wave supposed already established. For simplicity we will suppose p and p to be oonnected by Boyle's Law P = c*P (1) If. we write y = a; -f £, so that y denotes the absolute oo-oidinate at time t of the particle whose undisturbed abscissa is x, the equation (3) of Art. 281 becomes S-3/©' m This is satisfied by & " ■^ (X) ('' -^ M2)}"-/(2)' '« Hence a first integral of (2) is ^ = C' ± c log ^ (6) ot ox To obtain the * general integral' of (5) we must eliminate a between the equations y = or + (C± c log a) « + (a)\ = or ± c/ + a<^' (a), J ^"^' where ^ is arbitrary. Now ^ = ^, so that, if u be the velocity of the particle x, we have « = ^ = C±clog^« (7) 01 p On the outskirts of the wave we shall have u = 0, p = pq. It follows that C = 0, and therefore P=Po«'^'''' (8) * "On the Mathematical Theoiy of Sound," PkU. Trans, t. cl. p. 133 (1868). t See Forsyth, Differential Equations, c. iz. 282-284] Waves of Finite Amplitude 479 Hence in a pn^ressive wave p and u must be connected by this relation. If this be satisfied initially^ the function ^ which occurs in (6) is to be determined from the conditions at time ^ = by the equation <t>'{polp)= -x (9) To obtain results independent of the particular form of the wave, consider two particles (which we will distinguish by sufi&xee) so related that the value of p which obtains for the first particle at time t^ is found at the second particle at time t^. The value of a (= pjp) is the same for both, and therefore by (6), with C = 0, Vi- yi = «(«8 - a:i) ± c (t^ - t^) log a,\ The latter equation may be written -— = T c — , (11) ^ Pq shewing that the value of p is propagated from particle to particle at the rate p/pf, , c. The rate of propagation in apace is given by ^= Tc±c log a= qPc + M (12) This is in agreement with Riemann*s results, since on the present hypothesis of isothermal expansion (dpjdpY = c. For a wave travelling in the positive direction we must take the lower signs. If it be one of condensation (p > po), u is positive, by (8). It follows that the denser pckrts of the wave are continually gaining on the rarer, and at length overtake them ; the subsequent motion is then beyond the scope of our sknalysis. Eliminating x between the equations (6). and writing for c log a its value -t^ we find, lor a wave travelling in the positive direction, y=(c + u)t + F (a), (13) where J* is an arbitrary function. In virtue of (8) this is equivalent to n = f{y-(c + u)t} (U) This formula is due to Poisson*. Its interpretation, leading to the same results as above, for the mode of alteration of the wave as it proceeds, forms the subject of a paper by Stokes t. 284. The conditions for a wave of permanent type have been investigated in a very simple manner by Bankine|. Let Ay B he two points of an ideal tube of unit section drawn in the direction of propagation, whicb is (say) that of x positive, and let the values of the pressure, the density, and the particle-velocity at A and B be denoted by p-^, pi, u^ and p^, p^, u^, respectively. If, as in Art. 175, we impress on everything a velocity c equal and opposite to that of the wave, we reduce the problem to one of steady motion. Since * "M^moire sur la th^orie du son,'* Joum. de VicoU Polytechn, t. viL p. 367 (1807). t "On a Difficulty in the Theory of Sound," PhU, Mag. (3), t. xxiii. p. 349 (1848) [Paptrs, t. ii. p. 51]. } "On the Thermodynamic Theory of Waves of Finite Longitudinal Disturbance," Phil, Trans, t. clx. p. 277 (1870) [Papers, p. 630]. 480 Waves of Eoqparmon [ohap. x the same amount of matter now crosses in unit time each section of the tube, we have Pl (C - Ui) = Pj, (C - Wj) = w (1) say ; where m denotes the mass swept past in unit time by a plane moving with the wave, in the original form of the problem. This quantity m is called by Bankine the 'mass- velocity' of the wave. Again, the total force acting on the mass included between A and B is p^ - p^, in the direction BA, and the rate at which this mass is gaining momentum in the same direction is in(c — Uj) — m(c — u^). Hence P2 "" ?i = ^ (^t "" ^1) (2) Combined with (1) this gives l>i + — = P« +— (3) Pi P2 Hence a wave of finite amplitude could not be propagated unchanged except in a medium such that p-\ = const (4) This conclusion has already been arrived at, in a different manner, in Art. 281. It may be noticed that, if we write v = 1/p, the relation (4) is represented on Watt's, diagram by a straight line. If the variation of density be slight, the relation (4) may, however, be regarded as holding approximately for actual fluids, provided tn have the proper value. Putting p = Po (! + «)» P = Po'^KS, m=^p^c (5) we find c* = — , (6) - />o as in Art. 277. The fact that in actual fluids a progressive wave of finite amplitude continually alters its type, so that the variations of density towards the front become more and more abrupt, has led